The one-dimensional (1D) Schrödinger operator with Gaussian white noise is one of the few disordered spectral problems that can be treated nonperturbatively. The exact integrated density of states (IDOS), expressible in terms of Airy functions, follows from the pioneering work of Frisch and Lloyd and Halperin [1 , 2 ] . The IDOS and the Lyapunov exponent are encoded in the imaginary and real parts, respectively, of a complex characteristic exponent, related through the Herbert–Jones–Thouless relation [3 , 4 ] . We refer to this quantity as the Halperin–Thouless exponent . Its real part determines the inverse localization length [5 , 6 ] , a fundamental scale governing transport and eigenfunction statistics in 1D [7 , 8 , 9 , 10 , 11 ] .
Rigorous work has further developed a detailed probabilistic picture of localized eigenfunctions and spectral fluctuations [12 , 13 , 14 , 15 , 16 ] . Since strictly 1D systems are highly idealized, a natural question is how much of this solvability survives for an N N N N = 1 N = 1 N = 1 N N N
Multichannel localization is usually approached through the DMPK transport equation [17 , 18 ] . In the unitary class the DMPK equation is known to admit a mapping to free fermions [19 , 20 ] and is therefore exactly solvable for any length L L L N N N [21 ] . At the same time, the microscopic spectral problem for the multichannel disordered Hamiltonian H L \mathcal{H}_L H L [22 ] , apart from its infinite-channel limit equivalent to the supersymmetric nonlinear sigma model [25 , 26 ] .
In the last decade, the study of random matrix Schrödinger operators has also been motivated by their relation to Hessians at equilibria of directed elastic manifolds in random potentials and in Kac–Rice calculations of their complexity [27 , 28 , 29 ] . In this context, Fyodorov and Le Doussal obtained the shape of the spectral density in the limit N → ∞ N \to\infty N → ∞ [28 ] . More recently Fyodorov, Le Doussal, and Ossipov Fyodorov, Le Doussal, and Ossipov [30 ] studied the statistics of the spectral determinant det ( H L − E ) \det(\mathcal{H}_L-E) det ( H L − E ) N ≫ 1 N \gg1 N ≫ 1 [31 , 32 , 33 ] and a matrix Riccati formulation. This approach revealed a striking connection between the matrix Schrödinger problem and the problem of interacting particles in nonconfining potentials studied earlier by Allez and Dumaz [34 ] , who showed that a large-N N N N N N E ∗ E^* E ∗ [30 ] further estimated the finite-N N N E ∗ E^* E ∗ N = ∞ N = \infty N = ∞ ∝ N ( − E ) 3 / 2 \propto N(-E)^{3/2} ∝ N ( − E ) 3/2 ∝ N ( E ∗ − E ) 5 / 4 \propto N(E^* - E)^{5/4} ∝ N ( E ∗ − E ) 5/4 ∣ E − E ∗ ∣ ∝ N − 4 / 5 |E - E^*| \propto N^{-4/5} ∣ E − E ∗ ∣ ∝ N − 4/5 [35 , 36 , 37 , 38 ] , and to mesoscopic SNS junctions, where sigma-model instantons yield the same 5 / 4 5/4 5/4 [39 ] .
Consider the spectral problem for the following N × N N \times N N × N H L \mathcal{H}_L H L N N N ψ ( x ) \psi(x) ψ ( x ) x ∈ [ 0 , L ] x \in[0,L] x ∈ [ 0 , L ]
H L = − ∂ x 2 I N + W ( x ) , H L ψ = E ψ (1) \mathcal{H}_L=-\partial_x^2 I_N+W(x),\qquad\mathcal{H}_L\psi=E\psi \tag*{(1)} H L = − ∂ x 2 I N + W ( x ) , H L ψ = E ψ (1) where ψ \psi ψ ψ ( 0 ) = ψ ( L ) = 0 \psi(0)=\psi(L)=0 ψ ( 0 ) = ψ ( L ) = 0 I N I_N I N N × N N \times N N × N N × N N \times N N × N W ( x ) = W † ( x ) W(x)=W^\dagger(x) W ( x ) = W † ( x ) ⟨ W i j ( x ) ⟩ = 0 \langle W_{ij}(x)\rangle=0 ⟨ W ij ( x )⟩ = 0 ⟨ W i j ( x ) W k l ( x ′ ) ⟩ = 2 T δ i l δ j k δ ( x − x ′ ) \langle W_{ij}(x)W_{kl}(x')\rangle=2T\delta_{il}\delta_{jk}\delta(x-x') ⟨ W ij ( x ) W k l ( x ′ )⟩ = 2 T δ i l δ j k δ ( x − x ′ ) T = J 2 / ( 2 N ) T=J^2/(2N) T = J 2 / ( 2 N ) Y Y Y (1) : − Y ′ ′ + W Y = E Y -Y''+WY=EY − Y ′′ + W Y = E Y Y ( 0 ) = 0 Y(0)=0 Y ( 0 ) = 0 Y ′ ( 0 ) = I N Y'(0)=I_N Y ′ ( 0 ) = I N Z : = Y ′ Y − 1 Z:=Y'Y^{-1} Z := Y ′ Y − 1 H L \mathcal{H}_L H L
Z ′ = − E I N − Z 2 + W . (2) Z'=-EI_N-Z^2+W. \tag*{(2)} Z ′ = − E I N − Z 2 + W . (2) The eigenvalues λ i \lambda_i λ i Z Z Z
d λ i = [ − ( E + λ i 2 ) + 2 T ∑ j ≠ i 1 λ i − λ j ] d x + 2 T d B i , (3) \mathrm{d}\lambda_i=\left[-(E+\lambda_i^2)+2T\sum_{j\ne i}\frac{1}{\lambda_i-\lambda_j}\right]\mathrm{d}x+\sqrt{2T}\,\mathrm{d}B_i, \tag*{(3)} d λ i = − ( E + λ i 2 ) + 2 T j = i ∑ λ i − λ j 1 d x + 2 T d B i , (3) where the B i B_i B i [42 ] driven by the nonconfining cubic potential U E ( λ ) = E λ + λ 3 3 U_E(\lambda)=E\lambda+\frac{\lambda^3}{3} U E ( λ ) = E λ + 3 λ 3 P x ( λ ) P_x(\lambda) P x ( λ ) C N = { λ 1 < λ 2 < ⋯ < λ N } C_N=\{\lambda_1<\lambda_2<\cdots<\lambda_N\} C N = { λ 1 < λ 2 < ⋯ < λ N } ∂ x P x ( λ ) = L N F P P x ( λ ) = ∑ i = 1 N ∂ i J i [ P x ] ( λ ) \partial_xP_x(\lambda)=\mathcal{L}^{\mathrm{FP}}_NP_x(\lambda)=\sum_{i=1}^N\partial_i\mathcal{J}_i[P_x](\lambda) ∂ x P x ( λ ) = L N FP P x ( λ ) = ∑ i = 1 N ∂ i J i [ P x ] ( λ ) [43 ]
J i [ P ] ( λ ) : = T ∂ i P ( λ ) + [ E + λ i 2 − 2 T ∑ j ≠ i 1 λ i − λ j ] P ( λ ) , (4) \mathcal{J}_i[P](\lambda):=T\partial_iP(\lambda)+\left[E+\lambda_i^2-2T\sum_{j\ne i}\frac{1}{\lambda_i-\lambda_j}\right]P(\lambda), \tag*{(4)} J i [ P ] ( λ ) := T ∂ i P ( λ ) + E + λ i 2 − 2 T j = i ∑ λ i − λ j 1 P ( λ ) , (4) with ∂ i : = ∂ ∂ λ i \partial_i:=\frac{\partial}{\partial\lambda_i} ∂ i := ∂ λ i ∂ λ i = κ cot θ i \lambda_i=\kappa\cot\theta_i λ i = κ cot θ i θ i ≡ θ i + π \theta_i\equiv\theta_i+\pi θ i ≡ θ i + π λ i = − ∞ \lambda_i=-\infty λ i = − ∞ λ i = + ∞ \lambda_i=+\infty λ i = + ∞ ∣ d θ i d λ i ∣ = κ λ i 2 + κ 2 ∼ κ λ i 2 \left|\frac{\mathrm{d}\theta_i}{\mathrm{d}\lambda_i}\right|=\frac{\kappa}{\lambda_i^2+\kappa^2}\sim\frac{\kappa}{\lambda_i^2} d λ i d θ i = λ i 2 + κ 2 κ ∼ λ i 2 κ [44 ] shows that a density smooth in θ i \theta_i θ i λ i − 2 \lambda_i^{-2} λ i − 2 N > 1 N>1 N > 1 − ∞ -\infty − ∞ + ∞ +\infty + ∞
labels undergo a cyclic permutation. This cyclic projective gluing is the global boundary condition obeyed by the determinantal solution constructed below, see [45 ] . Matrix oscillation theory identifies the total number of Riccati windings with the number N L ( E ) : = # { E α < E } \mathcal{N}_L(E):=\#\{E_\alpha<E\} N L ( E ) := # { E α < E } E E E [46 , 47 , 48 ] . Each winding contributes one level; hence the stationary probability flux through the cyclic projective boundary is the IDOS per unit length:
ν N ( E ) : = lim L → ∞ 1 L ⟨ N L ( E ) ⟩ = J s t w i n d , ρ N ( E ) = ∂ E ν N ( E ) . (5) \nu_N(E):=\lim_{L\to\infty}\frac{1}{L}\langle\mathcal{N}_L(E)\rangle=\mathcal{J}_{\mathrm{st}}^{\mathrm{wind}},\qquad\rho_N(E)=\partial_E\nu_N(E). \tag*{(5)} ν N ( E ) := L → ∞ lim L 1 ⟨ N L ( E )⟩ = J st wind , ρ N ( E ) = ∂ E ν N ( E ) . (5) Here J i , s t : = J i [ P s t ] \mathcal{J}_{i,\mathrm{st}}:=\mathcal{J}_i[P_{\mathrm{st}}] J i , st := J i [ P st ] [49 ]
J s t w i n d : = ∫ λ 2 < ⋯ < λ N J 1 , s t ( − ∞ , λ 2 , … , λ N ) ∏ j = 2 N d λ j . (6) \mathcal{J}_{\mathrm{st}}^{\mathrm{wind}}:=\int_{\lambda_2<\cdots<\lambda_N}\mathcal{J}_{1,\mathrm{st}}(-\infty,\lambda_2,\ldots,\lambda_N)\prod_{j=2}^N\mathrm{d}\lambda_j. \tag*{(6)} J st wind := ∫ λ 2 < ⋯ < λ N J 1 , st ( − ∞ , λ 2 , … , λ N ) j = 2 ∏ N d λ j . (6) We now exploit a special property of (4) . Writing, in C N C_N C N P x ( λ ) = V N ( λ ) Ψ x ( λ ) P_x(\lambda)=V_N(\lambda)\Psi_x(\lambda) P x ( λ ) = V N ( λ ) Ψ x ( λ ) V N ( λ ) = ∏ i < j ( λ j − λ i ) V_N(\lambda)=\prod_{i<j}(\lambda_j-\lambda_i) V N ( λ ) = ∏ i < j ( λ j − λ i ) β = 2 \beta=2 β = 2
∑ i < j − ( E + λ i 2 ) + ( E + λ j 2 ) λ i − λ j = − ( N − 1 ) ∑ i λ i , (7) \sum_{i<j}\frac{-(E+\lambda_i^2)+(E+\lambda_j^2)}{\lambda_i-\lambda_j}=-(N-1)\sum_i\lambda_i, \tag*{(7)} i < j ∑ λ i − λ j − ( E + λ i 2 ) + ( E + λ j 2 ) = − ( N − 1 ) i ∑ λ i , (7) so the residual term collapses exactly to a one-body contribution. One finds
L N F P [ V N ( λ ) Ψ x ( λ ) ] = V N ( λ ) ∑ i = 1 N ℓ N ( i ) Ψ x ( λ ) , (8) \mathcal{L}^{\mathrm{FP}}_N[V_N(\lambda)\Psi_x(\lambda)]=V_N(\lambda)\sum_{i=1}^N\ell_N^{(i)}\Psi_x(\lambda), \tag*{(8)} L N FP [ V N ( λ ) Ψ x ( λ )] = V N ( λ ) i = 1 ∑ N ℓ N ( i ) Ψ x ( λ ) , (8) Upon continuation between ordering chambers, Ψ x = P x / V N \Psi_x=P_x/V_N Ψ x = P x / V N N N N ℓ N ( i ) \ell_N^{(i)} ℓ N ( i ) ℓ N \ell_N ℓ N λ i \lambda_i λ i
ℓ N = T ∂ λ 2 + ( E + λ 2 ) ∂ λ + ( N + 1 ) λ , \ell_N=T\partial_\lambda^2+(E+\lambda^2)\partial_\lambda+(N+1)\lambda, ℓ N = T ∂ λ 2 + ( E + λ 2 ) ∂ λ + ( N + 1 ) λ , ℓ N † = T ∂ λ 2 − ( E + λ 2 ) ∂ λ + ( N − 1 ) λ . (9) \ell_N^\dagger=T\partial_\lambda^2-(E+\lambda^2)\partial_\lambda+(N-1)\lambda. \tag*{(9)} ℓ N † = T ∂ λ 2 − ( E + λ 2 ) ∂ λ + ( N − 1 ) λ . (9) This second cancellation, rather than β = 2 \beta=2 β = 2 − ∞ -\infty − ∞ + ∞ +\infty + ∞ ( − 1 ) N − 1 (-1)^{N-1} ( − 1 ) N − 1 [45 ] .
The key finite-N N N ℓ N † \ell_N^\dagger ℓ N † N N N P N − 1 = span { 1 , λ , … , λ N − 1 } \mathcal{P}_{N-1}=\operatorname{span}\{1,\lambda,\ldots,\lambda^{N-1}\} P N − 1 = span { 1 , λ , … , λ N − 1 }
ℓ N † λ n = T n ( n − 1 ) λ n − 2 − E n λ n − 1 + ( N − 1 − n ) λ n + 1 , (10) \ell_N^\dagger\lambda^n=Tn(n-1)\lambda^{n-2}-En\lambda^{n-1}+(N-1-n)\lambda^{n+1}, \tag*{(10)} ℓ N † λ n = T n ( n − 1 ) λ n − 2 − E n λ n − 1 + ( N − 1 − n ) λ n + 1 , (10) so the raising term vanishes at n = N − 1 n=N-1 n = N − 1 λ n \lambda^n λ n ℓ N † ∣ P N − 1 \ell_N^\dagger|_{\mathcal{P}_{N-1}} ℓ N † ∣ P N − 1 Tr P N − 1 ℓ N † = 0 \operatorname{Tr}_{\mathcal{P}_{N-1}}\ell_N^\dagger=0 Tr P N − 1 ℓ N † = 0 N N N
ℓ N † p a = ϵ a p a , p a ∈ P N − 1 , a = 1 , … , N , (11) \ell_N^\dagger p_a=\epsilon_a p_a,\qquad p_a\in\mathcal{P}_{N-1},\qquad a=1,\ldots,N, \tag*{(11)} ℓ N † p a = ϵ a p a , p a ∈ P N − 1 , a = 1 , … , N , (11) whose eigenvalues satisfy ∑ a = 1 N ϵ a = 0 \sum_{a=1}^{N}\epsilon_a=0 ∑ a = 1 N ϵ a = 0 v a v_a v a
ℓ N v a = ϵ a v a . (12) \ell_N v_a=\epsilon_a v_a. \tag*{(12)} ℓ N v a = ϵ a v a . (12) Their Slater determinant then obeys
( ∑ i = 1 N ℓ N ( i ) ) det a , i [ v a ( λ i ) ] = ( ∑ a = 1 N ϵ a ) det a , i [ v a ( λ i ) ] = 0. (13) \left(\sum_{i=1}^{N}\ell_N^{(i)}\right)\det_{a,i}[v_a(\lambda_i)]=\left(\sum_{a=1}^{N}\epsilon_a\right)\det_{a,i}[v_a(\lambda_i)]=0. \tag*{(13)} ( i = 1 ∑ N ℓ N ( i ) ) a , i det [ v a ( λ i )] = ( a = 1 ∑ N ϵ a ) a , i det [ v a ( λ i )] = 0. (13) Since the p a p_a p a P N − 1 \mathcal{P}_{N-1} P N − 1 det a , i [ p a ( λ i ) ] ∝ V N ( λ ) \det_{a,i}[p_a(\lambda_i)]\propto V_N(\lambda) det a , i [ p a ( λ i )] ∝ V N ( λ ) (8) and (13) the stationary density takes the biorthogonal form [50 ]
P s t ( λ ; E ) = 1 Z N det a , i [ p a ( λ i ) ] det a , i [ v a ( λ i ) ] . (14) P_{\mathrm{st}}(\lambda;E)=\frac{1}{Z_N}\det_{a,i}[p_a(\lambda_i)]\det_{a,i}[v_a(\lambda_i)]. \tag*{(14)} P st ( λ ; E ) = Z N 1 a , i det [ p a ( λ i )] a , i det [ v a ( λ i )] . (14) For each polynomial left mode p a p_a p a v a v_a v a (12) up to normalization. We fix the normalization by setting the Lagrange concomitant
B [ p a , v a ] ≡ T ( p a v a ′ − p a ′ v a ) + ( E + λ 2 ) p a v a = 1. (15) \mathcal{B}[p_a,v_a]\equiv T(p_a v_a'-p_a'v_a)+(E+\lambda^2)p_av_a=1. \tag*{(15)} B [ p a , v a ] ≡ T ( p a v a ′ − p a ′ v a ) + ( E + λ 2 ) p a v a = 1. (15) The Lagrange identity p ℓ N v − v ℓ N † p = ∂ λ B [ p , v ] p\ell_Nv-v\ell_N^\dagger p=\partial_\lambda\mathcal{B}[p,v] p ℓ N v − v ℓ N † p = ∂ λ B [ p , v ] B [ p a , v a ] \mathcal{B}[p_a,v_a] B [ p a , v a ] λ \lambda λ p a ( λ ) v a ( λ ) ∼ ( E + λ 2 ) − 1 ∼ λ − 2 p_a(\lambda)v_a(\lambda)\sim(E+\lambda^2)^{-1}\sim\lambda^{-2} p a ( λ ) v a ( λ ) ∼ ( E + λ 2 ) − 1 ∼ λ − 2 [51 ]
v a ( λ ) = 1 T e − U E ( λ ) / T p a ( λ ) FP ∫ − ∞ λ e U E ( λ ′ ) / T p a ( λ ′ ) 2 d λ ′ . (16) v_a(\lambda)=\frac{1}{T}e^{-U_E(\lambda)/T}p_a(\lambda)\operatorname{FP}\int_{-\infty}^{\lambda}\frac{e^{U_E(\lambda')/T}}{p_a(\lambda')^2}\,\mathrm{d}\lambda'. \tag*{(16)} v a ( λ ) = T 1 e − U E ( λ ) / T p a ( λ ) FP ∫ − ∞ λ p a ( λ ′ ) 2 e U E ( λ ′ ) / T d λ ′ . (16) For N = 1 N=1 N = 1 p 1 = 1 p_1=1 p 1 = 1 (16) is Halperin’s constant-current solution normalized to unit current.
Both the IDOS and the total volume growth Γ N ( E ) : = lim L → ∞ L − 1 log ∣ det Y ( L ) ∣ = ∑ j γ j ( E ) \Gamma_N(E):=\lim_{L\to\infty}L^{-1}\log|\det Y(L)|=\sum_j\gamma_j(E) Γ N ( E ) := lim L → ∞ L − 1 log ∣ det Y ( L ) ∣ = ∑ j γ j ( E ) n ( λ ) : = ⟨ ∑ j δ ( λ − λ j ) ⟩ s t n(\lambda):=\langle\sum_j\delta(\lambda-\lambda_j)\rangle_{\mathrm{st}} n ( λ ) := ⟨ ∑ j δ ( λ − λ j ) ⟩ st N N N λ 1 → − ∞ \lambda_1\to-\infty λ 1 → − ∞ (4) , J 1 , s t ∼ λ 1 2 P s t \mathcal{J}_{1,\mathrm{st}}\sim\lambda_1^2P_{\mathrm{st}} J 1 , st ∼ λ 1 2 P st λ 1 \lambda_1 λ 1 n ( λ ) n(\lambda) n ( λ ) N − 1 N-1 N − 1 J s t w i n d ∼ λ 2 n ( λ ) \mathcal{J}_{\mathrm{st}}^{\mathrm{wind}}\sim\lambda^2n(\lambda) J st wind ∼ λ 2 n ( λ ) (5) , n ( λ ) ∼ ν N ( E ) / λ 2 n(\lambda)\sim\nu_N(E)/\lambda^2 n ( λ ) ∼ ν N ( E ) / λ 2 λ → + ∞ \lambda\to+\infty λ → + ∞ ∂ x log det Y = Tr Z \partial_x\log\det Y=\operatorname{Tr}Z ∂ x log det Y = Tr Z Γ N ( E ) = PV ∫ R λ n ( λ ) d λ \Gamma_N(E)=\operatorname{PV}\int_{\mathbb{R}}\lambda n(\lambda)\,\mathrm{d}\lambda Γ N ( E ) = PV ∫ R λn ( λ ) d λ s = − E / T 2 / 3 s=-E/T^{2/3} s = − E / T 2/3 q ≥ 0 q\geq0 q ≥ 0
F N ( q ; s ) : = ∫ R N P s t ( λ ; E ) exp [ − i q T 1 / 3 ∑ j = 1 N λ j ] d N λ , (17) \mathcal{F}_N(q;s):=\int_{\mathbb{R}^N}P_{\mathrm{st}}(\lambda;E)\exp\left[-\frac{\mathrm{i}q}{T^{1/3}}\sum_{j=1}^{N}\lambda_j\right]\mathrm{d}^N\lambda, \tag*{(17)} F N ( q ; s ) := ∫ R N P st ( λ ; E ) exp [ − T 1/3 i q j = 1 ∑ N λ j ] d N λ , (17) normalized by F N ( 0 ; s ) = 1 \mathcal{F}_N(0;s)=1 F N ( 0 ; s ) = 1 P s t P_{\mathrm{st}} P st R N \mathbb{R}^N R N q q q
F N ( q ; s ) = 1 − q T 1 / 3 [ π ν N ( E ) + i Γ N ( E ) ] + o ( q ) , q → 0 + . (18) \mathcal{F}_N(q;s)=1-\frac{q}{T^{1/3}}\left[\pi\nu_N(E)+\mathrm{i}\Gamma_N(E)\right]+o(q),\qquad q\to0^+. \tag*{(18)} F N ( q ; s ) = 1 − T 1/3 q [ π ν N ( E ) + i Γ N ( E ) ] + o ( q ) , q → 0 + . (18) Indeed, the leading small-q q q
1 − F N ( q ; s ) ≃ ∫ R n ( λ ) ( 1 − e − i q λ / T 1 / 3 ) d λ . (19) 1-\mathcal{F}_N(q;s)\simeq\int_{\mathbb{R}}n(\lambda)\left(1-e^{-\mathrm{i}q\lambda/T^{1/3}}\right)\,\mathrm{d}\lambda. \tag*{(19)} 1 − F N ( q ; s ) ≃ ∫ R n ( λ ) ( 1 − e − i q λ / T 1/3 ) d λ . (19) Using n ( λ ) ∼ ν N ( E ) / λ 2 n(\lambda)\sim\nu_N(E)/\lambda^2 n ( λ ) ∼ ν N ( E ) / λ 2 π ν N ( E ) q / T 1 / 3 \pi\nu_N(E)q/T^{1/3} π ν N ( E ) q / T 1/3 ∫ R ( 1 − cos u ) u − 2 d u = π \int_{\mathbb{R}}(1-\cos u)u^{-2}\,\mathrm{d}u=\pi ∫ R ( 1 − cos u ) u − 2 d u = π + ∞ +\infty + ∞ − ∞ -\infty − ∞ i q Γ N ( E ) / T 1 / 3 \mathrm{i}q\Gamma_N(E)/T^{1/3} i q Γ N ( E ) / T 1/3
Inserting the biorthogonal form (14) of P s t P_{\mathrm{st}} P st (17) Andréief’s identity reduces the N N N
F N ( q ; s ) = det I ( q ) det I ( 0 ) , I a b ( q ) = ∫ R p a ( x ) v b ( x ) e − i q x d x , (20) \mathcal{F}_N(q;s)=\frac{\det I(q)}{\det I(0)},\qquad I_{ab}(q)=\int_{\mathbb{R}}p_a(x)v_b(x)e^{-\mathrm{i}qx}\,\mathrm{d}x, \tag*{(20)} F N ( q ; s ) = det I ( 0 ) det I ( q ) , I ab ( q ) = ∫ R p a ( x ) v b ( x ) e − i q x d x , (20) where x = λ / T 1 / 3 x=\lambda/T^{1/3} x = λ / T 1/3 det I ( q ) \det I(q) det I ( q ) [45 ] , giving
F N ( q ; s ) = τ N ( s − i q ) τ N ( s ) , q > 0 , (21) \mathcal{F}_N(q;s)=\frac{\tau_N(s-\mathrm{i}q)}{\tau_N(s)},\qquad q>0, \tag*{(21)} F N ( q ; s ) = τ N ( s ) τ N ( s − i q ) , q > 0 , (21) in terms of the Hankel determinant
τ N ( z ) = det 0 ≤ j , k ≤ N − 1 [ A ( j + k ) ( z ) ] . (22) \tau_N(z)=\det_{0\leq j,k\leq N-1}\left[\mathcal{A}^{(j+k)}(z)\right]. \tag*{(22)} τ N ( z ) = 0 ≤ j , k ≤ N − 1 det [ A ( j + k ) ( z ) ] . (22) where A − ( z ) : = Ai ( z ) − i Bi ( z ) \mathcal{A}_-(z):=\operatorname{Ai}(z)-\mathrm{i}\operatorname{Bi}(z) A − ( z ) := Ai ( z ) − i Bi ( z ) A − ( m ) ( z ) : = ∂ z m A − ( z ) \mathcal{A}_-^{(m)}(z):=\partial_z^m\mathcal{A}_-(z) A − ( m ) ( z ) := ∂ z m A − ( z ) (21) gives F N ( q ; s ) = 1 − i q ∂ s log τ N ( s ) + o ( q ) \mathcal{F}_N(q;s)=1-\mathrm{i}q\,\partial_s\log\tau_N(s)+o(q) F N ( q ; s ) = 1 − i q ∂ s log τ N ( s ) + o ( q ) (18) we thus obtain the exact multichannel analogue of Halperin–Thouless complex characteristic exponent for any N N N
Ξ N ( E ) : = Γ N ( E ) − i π ν N ( E ) = T 1 / 3 ∂ s log τ N ( s ) ∣ s = − E / T 2 / 3 , (23) \Xi_N(E):=\Gamma_N(E)-\mathrm{i}\pi\nu_N(E)=T^{1/3}\left.\partial_s\log\tau_N(s)\right|_{s=-E/T^{2/3}}, \tag*{(23)} Ξ N ( E ) := Γ N ( E ) − i π ν N ( E ) = T 1/3 ∂ s log τ N ( s ) ∣ s = − E / T 2/3 , (23) From it one extracts the eigenvalue density ρ N ( E ) = 1 π T 1 / 3 Im ∂ s 2 log τ N ( s ) ∣ s = − E / T 2 / 3 \rho_N(E)=\frac{1}{\pi T^{1/3}}\operatorname{Im}\left.\partial_s^2\log\tau_N(s)\right|_{s=-E/T^{2/3}} ρ N ( E ) = π T 1/3 1 Im ∂ s 2 log τ N ( s ) s = − E / T 2/3 N = 1 N=1 N = 1 τ 1 = A − \tau_1=\mathcal{A}_- τ 1 = A − [2 ] and the associated Lyapunov exponent [6 ] .
Using A − ( z ) = 2 e − i π / 3 Ai ( z e 2 π i / 3 ) \mathcal{A}_-(z)=2e^{-\mathrm{i}\pi/3}\operatorname{Ai}(ze^{2\pi\mathrm{i}/3}) A − ( z ) = 2 e − i π /3 Ai ( z e 2 π i /3 ) Ai \operatorname{Ai} Ai [52 ] ; see Eqs. (9.2.11) and (9.5.4), one obtains
A − ( z ) = i π ∫ C − exp ( w 3 3 − z w ) d w , (24) \mathcal{A}_-(z)=\frac{\mathrm{i}}{\pi}\int_{\mathcal{C}_-}\exp\left(\frac{w^3}{3}-zw\right)\,\mathrm{d}w, \tag*{(24)} A − ( z ) = π i ∫ C − exp ( 3 w 3 − z w ) d w , (24) where C − : e i π / 3 ∞ ⟶ e i π ∞ \mathcal{C}_{-}:e^{i\pi/3}\infty\longrightarrow e^{i\pi}\infty C − : e iπ /3 ∞ ⟶ e iπ ∞ w = − λ / T 1 / 3 w=-\lambda/T^{1/3} w = − λ / T 1/3 Γ − : e − 2 π i / 3 ∞ ⟶ + ∞ \Gamma_{-}:e^{-2\pi i/3}\infty\longrightarrow+\infty Γ − : e − 2 π i /3 ∞ ⟶ + ∞
Applying Andréief’s identity in reverse to the Hankel determinant (22) and using (24) gives a scalar-source version of the Kontsevich matrix Airy integral [53 ] on the contour Γ − \Gamma_{-} Γ − [54 ] ,
Z N ( − ) ( E ) : = 1 N ! ∫ Γ N − V N ( λ ) 2 ∏ j = 1 N e − N V E ( λ j ) d N λ = C N T N 2 / 3 τ N ( − E / T 2 / 3 ) , (25) \begin{aligned}
\mathcal{Z}_{N}^{(-)}(E):=\frac{1}{N!}\int_{\Gamma_{N}^{-}}V_{N}(\lambda)^{2}\prod_{j=1}^{N}e^{-NV_{E}(\lambda_{j})}\,\mathrm{d}^{N}\lambda\\
&=C_{N}T^{N^{2}/3}\tau_{N}\left(-E/T^{2/3}\right), \tag*{(25)}
\end{aligned} Z N ( − ) ( E ) := N ! 1 ∫ Γ N − V N ( λ ) 2 j = 1 ∏ N e − N V E ( λ j ) d N λ = C N T N 2 /3 τ N ( − E / T 2/3 ) , (25) with V E ( λ ) = 2 J 2 U E ( λ ) V_{E}(\lambda)=\frac{2}{J^{2}}U_{E}(\lambda) V E ( λ ) = J 2 2 U E ( λ ) C N C_{N} C N E E E [55 ] ; in particular, David emphasized the role of complex integration contours in defining the unstable cubic model nonperturbatively [56 ] . Here however, the contour is selected by the projective boundary condition of the Dyson–Riccati dynamics. We thus obtain a distinct microscopic physical realization of the cubic matrix integral, for which Ξ N ( E ) = − T ∂ E log Z N ( − ) ( E ) \Xi_{N}(E)=-T\partial_{E}\log\mathcal{Z}_{N}^{(-)}(E) Ξ N ( E ) = − T ∂ E log Z N ( − ) ( E ) ν N ( E ) \nu_{N}(E) ν N ( E ) Γ N ( E ) \Gamma_{N}(E) Γ N ( E )
Let us now obtain the large-∣ E ∣ |E| ∣ E ∣ N N N τ N ∝ ∫ C N V N 2 ( w ) ∏ j e ϕ ( w j ) d w j \tau_{N}\propto\int_{\mathcal{C}_{N}}V_{N}^{2}(w)\prod_{j}e^{\phi(w_{j})}\,\mathrm{d}w_{j} τ N ∝ ∫ C N V N 2 ( w ) ∏ j e ϕ ( w j ) d w j ϕ ( w ) = w 3 / 3 − s w \phi(w)=w^{3}/3-sw ϕ ( w ) = w 3 /3 − s w w ± = ± s w_{\pm}=\pm\sqrt{s} w ± = ± s C − \mathcal{C}_{-} C − w 0 = − s w_{0}=-\sqrt{s} w 0 = − s E < 0 E<0 E < 0 s > 0 s>0 s > 0 E > 0 E>0 E > 0 s < 0 s<0 s < 0 w 0 = i ∣ s ∣ w_{0}=i\sqrt{|s|} w 0 = i ∣ s ∣
log τ N ≃ N ϕ ( w 0 ) − N 2 2 log w 0 − N ( 4 N 2 + 1 ) 48 w 0 3 + ⋯ , (26) \log\tau_{N}\simeq N\phi(w_{0})-\frac{N^{2}}{2}\log w_{0}-\frac{N(4N^{2}+1)}{48w_{0}^{3}}+\cdots, \tag*{(26)} log τ N ≃ N ϕ ( w 0 ) − 2 N 2 log w 0 − 48 w 0 3 N ( 4 N 2 + 1 ) + ⋯ , (26) up to s s s
For E ≫ T 2 / 3 E\gg T^{2/3} E ≫ T 2/3 (26) yields
ρ N ( E ) N = 1 2 π E [ 1 − 5 ( 4 N 2 + 1 ) T 2 32 E 3 + ⋯ ] , (27) \frac{\rho_{N}(E)}{N}=\frac{1}{2\pi\sqrt{E}}\left[1-\frac{5(4N^{2}+1)T^{2}}{32E^{3}}+\cdots\right], \tag*{(27)} N ρ N ( E ) = 2 π E 1 [ 1 − 32 E 3 5 ( 4 N 2 + 1 ) T 2 + ⋯ ] , (27) and Γ N = N 2 T / 4 E + ⋯ \Gamma_{N}=N^{2}T/4E+\cdots Γ N = N 2 T /4 E + ⋯ − E ≫ T 2 / 3 -E\gg T^{2/3} − E ≫ T 2/3 w − w_{-} w − Γ N ≃ N − E \Gamma_{N}\simeq N\sqrt{-E} Γ N ≃ N − E w + w_{+} w + N − 1 N-1 N − 1
ν N ( E ) ≃ 2 3 N − 3 π ( N − 1 ) ! ( − E ) ( 3 N − 2 ) / 2 T N − 1 e − 4 ( − E ) 3 / 2 / 3 T [ 1 − 21 N 2 − 33 N + 17 24 T ( − E ) 3 / 2 + ⋯ ] . (28) \nu_{N}(E)\simeq\frac{2^{3N-3}}{\pi(N-1)!}\frac{(-E)^{(3N-2)/2}}{T^{N-1}}e^{-4(-E)^{3/2}/3T}\left[1-\frac{21N^{2}-33N+17}{24}\frac{T}{(-E)^{3/2}}+\cdots\right]. \tag*{(28)} ν N ( E ) ≃ π ( N − 1 )! 2 3 N − 3 T N − 1 ( − E ) ( 3 N − 2 ) /2 e − 4 ( − E ) 3/2 /3 T [ 1 − 24 21 N 2 − 33 N + 17 ( − E ) 3/2 T + ⋯ ] . (28) At fixed J J J 8 N ( − E ) 3 / 2 / 3 J 2 8N(-E)^{3/2}/3J^{2} 8 N ( − E ) 3/2 /3 J 2 O ( N ) O(N) O ( N ) N = 1 N=1 N = 1
Figure 1. Finite-channel density of states and the rounding of the collective edge, at fixed disorder J = 1 J=1 J = 1 T = J 2 / 2 N T=J^{2}/2N T = J 2 /2 N ρ N ( E ) / N \rho_{N}(E)/N ρ N ( E ) / N N = 1 , 2 , 3 , 5 N=1,2,3,5 N = 1 , 2 , 3 , 5 N = ∞ N=\infty N = ∞ [28 ] ; the vanishing of the associated cubic discriminant reproduces E ∗ = − 3 4 J 2 E^{*}=-\frac{3}{4}J^{2} E ∗ = − 4 3 J 2 ( 2 π E ) − 1 (2\pi\sqrt{E})^{-1} ( 2 π E ) − 1 (27) . (b) Logarithmic representation: at every finite N N N (28) , including its full N N N N = ∞ N=\infty N = ∞ E < E ∗ E<E^{*} E < E ∗ N N N N N N
to the Airy expansions. As N N N J J J E ∗ E^{*} E ∗
Near the edge we use the monic polynomials π n \pi_{n} π n Γ − \Gamma_{-} Γ − e − N V E e^{-NV_{E}} e − N V E [55 ] . Their three-term recurrence defines the coefficients β n , r n \beta_{n},r_{n} β n , r n λ π n = π n + 1 + β n π n + r n π n − 1 \lambda\pi_{n}=\pi_{n+1}+\beta_{n}\pi_{n}+r_{n}\pi_{n-1} λ π n = π n + 1 + β n π n + r n π n − 1 N N N Γ − \Gamma_{-} Γ −
β n 2 + r n + r n + 1 + E = 0 , \beta_{n}^{2}+r_{n}+r_{n+1}+E=0, β n 2 + r n + r n + 1 + E = 0 , r n ( β n + β n − 1 ) = n J 2 2 N . (29) r_{n}(\beta_{n}+\beta_{n-1})=\frac{nJ^{2}}{2N}. \tag*{(29)} r n ( β n + β n − 1 ) = 2 N n J 2 . (29) In the planar N ⟶ ∞ N\longrightarrow\infty N ⟶ ∞ t = n / N t=n/N t = n / N t t t β n , N → d ( t ) \beta_{n,N}\to d(t) β n , N → d ( t ) r n , N → r ( t ) r_{n,N}\to r(t) r n , N → r ( t ) n = N n=N n = N t = 1 t=1 t = 1 (29) then reduces to d 2 + 2 r + E = 0 d^2+2r+E=0 d 2 + 2 r + E = 0 2 d r = J 2 / 2 2dr=J^2/2 2 d r = J 2 /2 d 3 + E d + J 2 / 2 = 0 d^3+Ed+J^2/2=0 d 3 + E d + J 2 /2 = 0 3 d ∗ 2 + E ∗ = 0 3d_\ast^2+E^\ast=0 3 d ∗ 2 + E ∗ = 0 d ∗ = s J / 2 d_\ast=s_J/2 d ∗ = s J /2 E ∗ = − 3 s J 2 / 4 E^\ast=-3s_J^2/4 E ∗ = − 3 s J 2 /4 s J = ( 2 J 2 ) 1 / 3 s_J=(2J^2)^{1/3} s J = ( 2 J 2 ) 1/3 [28 , 30 ] and with the confined-to-flowing transition point of [34 ] . Near this point E − E ∗ ≃ − 3 ( d − d ∗ ) 2 E-E^\ast\simeq-3(d-d_\ast)^2 E − E ∗ ≃ − 3 ( d − d ∗ ) 2 d − d ∗ d-d_\ast d − d ∗ N N N
Near the double root, let ℓ \ell ℓ n − N = O ( ℓ ) n-N=O(\ell) n − N = O ( ℓ ) A A A β n , N − d ∗ = O ( s J A ) \beta_{n,N}-d_\ast=O(s_JA) β n , N − d ∗ = O ( s J A ) r n , N − r ∗ = O ( s J 2 A ) r_{n,N}-r_\ast=O(s_J^2A) r n , N − r ∗ = O ( s J 2 A ) A 2 A^2 A 2 n n n A / ℓ 2 A/\ell^2 A / ℓ 2 ( n − N ) / N (n-N)/N ( n − N ) / N ℓ / N \ell/N ℓ / N A 2 ∼ A / ℓ 2 ∼ ℓ / N A^2\sim A/\ell^2\sim\ell/N A 2 ∼ A / ℓ 2 ∼ ℓ / N ℓ ∼ N 1 / 5 \ell\sim N^{1/5} ℓ ∼ N 1/5 A ∼ N − 2 / 5 A\sim N^{-2/5} A ∼ N − 2/5 E − E ∗ = O ( s J 2 A 2 ) E-E^\ast=O(s_J^2A^2) E − E ∗ = O ( s J 2 A 2 ) δ = E − E ∗ ∼ N − 4 / 5 \delta=E-E^\ast\sim N^{-4/5} δ = E − E ∗ ∼ N − 4/5 L δ N δ 3 / 2 ∼ 1 L_\delta N\delta^{3/2}\sim1 L δ N δ 3/2 ∼ 1 L δ ∼ δ − 1 / 4 L_\delta\sim\delta^{-1/4} L δ ∼ δ − 1/4 δ ∼ N − 4 / 5 \delta\sim N^{-4/5} δ ∼ N − 4/5 [45 ] . In this double-scaling window Eq. (29) reduces to Painlevé I.
As shown in End Matter, the endpoint recurrence coefficient is related exactly to the spectral observables by ∂ E Ξ N = − r N , N / T \partial_E\Xi_N=-r_{N,N}/T ∂ E Ξ N = − r N , N / T ρ N = ( π T ) − 1 Im r N , N \rho_N=(\pi T)^{-1}\operatorname{Im}r_{N,N} ρ N = ( π T ) − 1 Im r N , N r N , N − r ∗ = O ( s J 2 N − 2 / 5 ) r_{N,N}-r_\ast=O(s_J^2N^{-2/5}) r N , N − r ∗ = O ( s J 2 N − 2/5 ) T = s J 2 / ( 4 N ) T=s_J^2/(4N) T = s J 2 / ( 4 N ) N 3 / 5 / s J N^{3/5}/s_J N 3/5 / s J E = E ∗ + s J 2 N − 4 / 5 X E=E^\ast+s_J^2N^{-4/5}X E = E ∗ + s J 2 N − 4/5 X
ρ N ( E ) ∼ N 3 / 5 s J Φ ′ ( X ) , Φ ′ ( X ) = − 2 4 / 5 π Im y 1 ( 2 7 / 5 X ) . (30) \rho_N(E)\sim\frac{N^{3/5}}{s_J}\Phi'(X),\qquad\Phi'(X)=-\frac{2^{4/5}}{\pi}\operatorname{Im}y_1(2^{7/5}X). \tag*{(30)} ρ N ( E ) ∼ s J N 3/5 Φ ′ ( X ) , Φ ′ ( X ) = − π 2 4/5 Im y 1 ( 2 7/5 X ) . (30) where y 1 ( t ) y_1(t) y 1 ( t ) [57 ] of Painlevé I equation y ′ ′ ( t ) = 6 y ( t ) 2 + t y''(t)=6y(t)^2+t y ′′ ( t ) = 6 y ( t ) 2 + t y 1 y_1 y 1 [57 , 58 ] (End Matter). This solution connects the density smoothly through the threshold E = E ∗ E=E^\ast E = E ∗ X → + ∞ X\to+\infty X → + ∞ N N N
Φ ′ ( X ) ∼ 2 π 3 X 1 / 2 , (31) \Phi'(X)\sim\frac{2}{\pi\sqrt{3}}X^{1/2}, \tag*{(31)} Φ ′ ( X ) ∼ π 3 2 X 1/2 , (31) while for X → − ∞ X\to-\infty X → − ∞ y 1 y_1 y 1 [59 ] :
log Φ ′ ( X ) ∼ − 16 5 2 3 1 / 4 ∣ X ∣ 5 / 4 , (32) \log\Phi'(X)\sim-\frac{16}{5}\sqrt{2}3^{1/4}|X|^{5/4}, \tag*{(32)} log Φ ′ ( X ) ∼ − 5 16 2 3 1/4 ∣ X ∣ 5/4 , (32) This exponent is the action of small fluctuations around the real algebraic branch of Painlevé I. In physical units it equals β / T \beta/T β / T β = 4 5 2 3 1 / 4 s J 1 / 2 ( E ∗ − E ) 5 / 4 \beta=\frac45\sqrt{2}3^{1/4}s_J^{1/2}(E^\ast-E)^{5/4} β = 5 4 2 3 1/4 s J 1/2 ( E ∗ − E ) 5/4 [30 ] (at β = 2 \beta=2 β = 2 [39 ] , whose deep-tail limit B → 4 3 ( − E ) 3 / 2 \mathcal{B}\to\frac43(-E)^{3/2} B → 3 4 ( − E ) 3/2 (28) .
The critical-window scale E − E ∗ ∼ N − 4 / 5 E-E^\ast\sim N^{-4/5} E − E ∗ ∼ N − 4/5 5 / 4 5/4 5/4 (32) arose earlier in studies of other classes of quasi-one-dimensional disordered models, such as random banded matrices (RBM), see [60 , 61 , 62 ] for early physics insights into their spectral bulk properties and [63 , 64 ] for recent mathematical progress. For an M × M M\times M M × M β = 2 \beta=2 β = 2 W W W 2 2 2 σ 2 ( ξ ) \sigma_2(\xi) σ 2 ( ξ ) σ 2 ( ξ ) = lim W 6 / M → 1 { # ( α : ( 2 − λ α ) W 4 / 5 < ξ ) } \sigma_2(\xi)=\lim_{W^6/M\to1}\{\#(\alpha:(2-\lambda_\alpha)W^{4/5}<\xi)\} σ 2 ( ξ ) = lim W 6 / M → 1 { # ( α : ( 2 − λ α ) W 4/5 < ξ )} [65 ] identified the corresponding edge crossover diagrammatically, and Sodin [66 ] subsequently proved the W − 4 / 5 W^{-4/5} W − 4/5 σ 2 ( ξ ) ∼ 2 ξ 3 / 2 / ( 3 π ) \sigma_2(\xi)\sim2\xi^{3/2}/(3\pi) σ 2 ( ξ ) ∼ 2 ξ 3/2 / ( 3 π ) ∣ ξ ∣ 5 / 4 |\xi|^{5/4} ∣ ξ ∣ 5/4 [67 , 68 ] for the modern ϕ 3 \phi^3 ϕ 3
We conjecture the one-point correspondence
Φ ( c ξ ) = A σ 2 ( ξ ) , c = ( 3 64 ) 1 / 5 , A = 48 − 1 / 5 , (33) \Phi(c\xi)=\mathcal{A}\sigma_2(\xi),\qquad c=\left(\frac{3}{64}\right)^{1/5},\qquad\mathcal{A}=48^{-1/5}, \tag*{(33)} Φ ( c ξ ) = A σ 2 ( ξ ) , c = ( 64 3 ) 1/5 , A = 4 8 − 1/5 , (33) where Φ ( X ) : = ∫ − ∞ X Φ ′ ( Y ) d Y \Phi(X):=\int_{-\infty}^{X}\Phi'(Y)\,\mathrm{d}Y Φ ( X ) := ∫ − ∞ X Φ ′ ( Y ) d Y ξ − 7 / 2 \xi^{-7/2} ξ − 7/2 (33) , the forbidden tail is therefore predicted sharply as − log σ 2 ( ξ ) ∼ ( 8 3 / ξ ) ∣ ξ ∣ 5 / 4 = 2.771281 … ∣ ξ ∣ 5 / 4 -\log\sigma_2(\xi)\sim(8\sqrt{3}/\xi)|\xi|^{5/4}=2.771281\ldots|\xi|^{5/4} − log σ 2 ( ξ ) ∼ ( 8 3 / ξ ) ∣ ξ ∣ 5/4 = 2.771281 … ∣ ξ ∣ 5/4 ξ → − ∞ \xi\to-\infty ξ → − ∞
Fermionization also applies to the finite-length evolution. Assign a factor z z z G L [ z ] ( λ , μ ) G^{[z]}_L(\lambda,\mu) G L [ z ] ( λ , μ ) ℓ N \ell_N ℓ N [45 ] . Matrix oscillation then gives the finite-length Dirichlet level-counting generating function via an ordered chamber integral:
F N ( z ; E , L ) : = ⟨ z N L ( E ) ⟩ = lim μ → + ∞ D ∫ C N V N ( λ ) V N ( μ ) × det i , j [ G L [ z ] ( λ i , μ j ) ] d N λ . (34) \mathcal{F}_N(z;E,L):=\left\langle z^{\mathcal{N}_L(E)}\right\rangle=\lim_{\mu\to+\infty}^{\mathrm{D}}\int_{\mathcal{C}_N}\frac{V_N(\lambda)}{V_N(\mu)}\times\det_{i,j}\left[G^{[z]}_L(\lambda_i,\mu_j)\right]\,\mathrm{d}^N\lambda. \tag*{(34)} F N ( z ; E , L ) := ⟨ z N L ( E ) ⟩ = μ → + ∞ lim D ∫ C N V N ( μ ) V N ( λ ) × i , j det [ G L [ z ] ( λ i , μ j ) ] d N λ . (34) Here lim μ → + ∞ D \lim_{\mu\to+\infty}^{\mathrm{D}} lim μ → + ∞ D Z ( x ) ≃ I N / x Z(x)\simeq I_N/x Z ( x ) ≃ I N / x z = e i χ z=e^{i\chi} z = e i χ (34) is the characteristic function of the level count. At z = 0 z=0 z = 0 F N ( 0 ; E , L ) = Pr { E 0 ( L ) > E } \mathcal{F}_N(0;E,L)=\Pr\{\mathcal{E}_0(L)>E\} F N ( 0 ; E , L ) = Pr { E 0 ( L ) > E } ℓ N \ell_N ℓ N N N N H N = − T ∂ λ 2 + ( E + λ 2 ) 2 / ( 4 T ) − N λ H_N=-T\partial_\lambda^2+(E+\lambda^2)^2/(4T)-N\lambda H N = − T ∂ λ 2 + ( E + λ 2 ) 2 / ( 4 T ) − N λ L L L [27 ] to the multichannel problem; see [45 ] .
In summary, the unitary matrix white-noise Schrödinger operator remains fully integrable for any length L L L N N N L → ∞ L\to\infty L → ∞ [69 ] reflects the fact that near the edge spectral counting is controlled by a localization volume rather than by the full system; see [45 ] . The same solution is simultaneously an exact finite-N N N [30 , 34 ] at β = 2 \beta=2 β = 2 [70 ] , while L → ∞ L\to\infty L → ∞ β = 1 , 4 \beta=1,4 β = 1 , 4 [71 ] .
Acknowledgments
Both authors acknowledge support from CNRS IRP-Hidings. YVF has been supported by grant UKRI1015, “Non-Hermitian random matrices: theory and applications.” and PLD by ANR Grant No. ANR-23-CE30-0020-01 EDIPS. The authors acknowledge the use of AI-based large language model assistants for help with asymptotic analysis, numerical computations and understanding the content of [67 ] . All scientific arguments, derivations, numerical results, and conclusions are the responsibility of the authors.
References
End Matter
We explain the main steps connecting the stationary Riccati solution to the Airy determinant and its asymptotic regimes. Detailed derivations are given in the Supplemental Material [45 ] .
A. From the stationary modes to the Airy determinant.— In the dimensionless variables x = λ / T 1 / 3 x=\lambda/T^{1/3} x = λ / T 1/3 s = − E / T 2 / 3 s=-E/T^{2/3} s = − E / T 2/3 F N ( q ; s ) = det I ( q ) / det I ( 0 ) \mathcal{F}_N(q;s)=\det I(q)/\det I(0) F N ( q ; s ) = det I ( q ) / det I ( 0 ) I a b ( q ) = ∫ R p a ( x ) v b ( x ) e − i q x d x I_{ab}(q)=\int_{\mathbb{R}}p_a(x)v_b(x)e^{-iqx}\,\mathrm{d}x I ab ( q ) = ∫ R p a ( x ) v b ( x ) e − i q x d x p a p_a p a N − 1 N-1 N − 1
Indeed, writing e b = ε b / T 1 / 3 e_b=\varepsilon_b/T^{1/3} e b = ε b / T 1/3 z = s − i q z=s-iq z = s − i q
[ ( z − s ) ( ∂ z 2 − z ) − ( N − 1 ) ∂ z + e b ] v ~ b = 0 , q > 0. \left[(z-s)(\partial_z^2-z)-(N-1)\partial_z+e_b\right]\widetilde v_b=0,\qquad q>0. [ ( z − s ) ( ∂ z 2 − z ) − ( N − 1 ) ∂ z + e b ] v b = 0 , q > 0. The contour transform T p = ∫ Γ − p ( x ) e − x 3 / 3 + z x d x \mathcal{T}p=\int_{\Gamma_-}p(x)e^{-x^3/3+zx}\,\mathrm{d}x T p = ∫ Γ − p ( x ) e − x 3 /3 + z x d x x x x ∂ z \partial_z ∂ z T 1 \mathcal{T}_1 T 1 A − = Ai − i Bi \mathcal{A}_-= \operatorname{Ai}-i\operatorname{Bi} A − = Ai − i Bi p b ( x ) = ∑ n < N c b n x n p_b(x)=\sum_{n<N}c_{bn}x^n p b ( x ) = ∑ n < N c bn x n
v ~ b ( q ) = d b ∑ n < N c b n A − ( n ) ( s − i q ) . \widetilde v_b(q)=d_b\sum_{n<N}c_{bn}\mathcal{A}_-^{(n)}(s-iq). v b ( q ) = d b n < N ∑ c bn A − ( n ) ( s − i q ) . Here the outgoing Airy combination is fixed by the physical boundary condition. The tails v b = O ( x − N − 1 ) v_b=O(x^{-N-1}) v b = O ( x − N − 1 ) v ~ b ( q ) → 0 \widetilde v_b(q)\to0 v b ( q ) → 0 q → + ∞ q\to+\infty q → + ∞ z = s − i q z=s-iq z = s − i q A − \mathcal{A}_- A − Ai + i Bi \operatorname{Ai}+i\operatorname{Bi} Ai + i Bi
Since multiplication by x x x i ∂ q = ∂ z i\partial_q=\partial_z i ∂ q = ∂ z I a b ( q ) = p a ( ∂ z ) v ~ b ( q ) I_{ab}(q)=p_a(\partial_z)\widetilde v_b(q) I ab ( q ) = p a ( ∂ z ) v b ( q ) v ~ b \widetilde v_b v b
I a b ( q ) = d b ∑ m , n < N c a m c b n A − ( m + n ) ( s − i q ) . I_{ab}(q)=d_b\sum_{m,n<N}c_{am}c_{bn}\mathcal{A}_-^{(m+n)}(s-iq). I ab ( q ) = d b m , n < N ∑ c am c bn A − ( m + n ) ( s − i q ) . Thus I ( q ) = C H ( s − i q ) C T D I(q)=CH(s-iq)C^{T}D I ( q ) = C H ( s − i q ) C T D C = ( c a n ) C=(c_{an}) C = ( c an ) D = diag ( d b ) D=\operatorname{diag}(d_b) D = diag ( d b ) H m n ( z ) = A − ( m + n ) ( z ) H_{mn}(z)=\mathcal{A}_-^{(m+n)}(z) H mn ( z ) = A − ( m + n ) ( z ) q q q (21) . Continuity supplies q = 0 q=0 q = 0
B. Fixed-N N N The cubic integral has phase φ ( w ) = w 3 / 3 − s w \varphi(w)=w^3/3-sw φ ( w ) = w 3 /3 − s w w ± = ± s w_\pm=\pm\sqrt{s} w ± = ± s w 0 w_0 w 0 w j = w 0 + ( − 2 w 0 ) − 1 / 2 h j w_j=w_0+(-2w_0)^{-1/2}h_j w j = w 0 + ( − 2 w 0 ) − 1/2 h j H H H e − Tr H 2 / 2 e^{-\operatorname{Tr}H^2/2} e − Tr H 2 /2 ⟨ ( Tr H 3 ) 2 ⟩ G U E = 3 N ( 4 N 2 + 1 ) \langle(\operatorname{Tr}H^3)^2\rangle_{\mathrm{GUE}}=3N(4N^2+1) ⟨( Tr H 3 ) 2 ⟩ GUE = 3 N ( 4 N 2 + 1 ) (26) . For E > 0 E>0 E > 0 w 0 = i ∣ s ∣ w_0=i\sqrt{|s|} w 0 = i ∣ s ∣ (27) .
For E < 0 E<0 E < 0 w − = − s w_-=-\sqrt{s} w − = − s s s s Γ N \Gamma_N Γ N w + = + s w_+=+\sqrt{s} w + = + s N − 1 N-1 N − 1 w − w_- w − e − 4 s 3 / 2 / 3 e^{-4s^{3/2}/3} e − 4 s 3/2 /3 i / 2 i/2 i /2 s s s N N N
τ N ( 1 ) τ N ( 0 ) ≃ i 2 3 N − 4 ( N − 1 ) ! s 3 ( N − 1 ) / 2 e − 4 s 3 / 3 . (35) \frac{\tau_N^{(1)}}{\tau_N^{(0)}} \simeq\frac{\mathrm{i}^{2^{3N-4}}}{(N-1)!}s^{3(N-1)/2}e^{-4s^3/3}. \tag*{(35)} τ N ( 0 ) τ N ( 1 ) ≃ ( N − 1 )! i 2 3 N − 4 s 3 ( N − 1 ) /2 e − 4 s 3 /3 . (35) Here the superscripts denote the one-variable and dominant saddle contributions. Using ν N = − ( T 1 / 3 / π ) Im ∂ s log τ N \nu_N=-(T^{1/3}/\pi)\operatorname{Im}\partial_s\log\tau_N ν N = − ( T 1/3 / π ) Im ∂ s log τ N (28) ; their evaluation and an independent norm-based derivation are given in [45 ] . This is a fixed-N N N
Recurrence coefficients and the collective edge
The monic polynomials π n \pi_n π n Γ − \Gamma_- Γ − ⟨ π n , π m ⟩ − = h n δ n m \langle\pi_n,\pi_m\rangle_-=h_n\delta_{nm} ⟨ π n , π m ⟩ − = h n δ nm
⟨ f , g ⟩ − : = ∫ Γ − f ( λ ) g ( λ ) e − N V E ( λ ) d λ \langle f,g\rangle_-:=\int_{\Gamma_-}f(\lambda)g(\lambda)e^{-NV_E(\lambda)}\,\mathrm{d}\lambda ⟨ f , g ⟩ − := ∫ Γ − f ( λ ) g ( λ ) e − N V E ( λ ) d λ without complex conjugation. Then r n = h n / h n − 1 r_n=h_n/h_{n-1} r n = h n / h n − 1 Z N ( − ) = ∏ n < N h n Z_N^{(-)}=\prod_{n<N}h_n Z N ( − ) = ∏ n < N h n π n 2 e − N V E \pi_n^2e^{-NV_E} π n 2 e − N V E π n π n − 1 e − N V E \pi_n\pi_{n-1}e^{-NV_E} π n π n − 1 e − N V E (29) ; boundary terms vanish in the cubic decay sectors.
To connect the recurrence problem to the physical density, set θ = − E / T \theta=-E/T θ = − E / T ∂ θ log h n = β n \partial_\theta\log h_n=\beta_n ∂ θ log h n = β n ∂ θ β n = r n + 1 − r n \partial_\theta\beta_n=r_{n+1}-r_n ∂ θ β n = r n + 1 − r n ∂ E 2 log Z N ( − ) = r N , N / T 2 \partial_E^2\log Z_N^{(-)}=r_{N,N}/T^2 ∂ E 2 log Z N ( − ) = r N , N / T 2
∂ E Ξ N = − r N , N T , ρ N ( E ) = Im r N , N π T . \partial_E\Xi_N=-\frac{r_{N,N}}{T},\qquad\rho_N(E)=\frac{\operatorname{Im}r_{N,N}}{\pi T}. ∂ E Ξ N = − T r N , N , ρ N ( E ) = π T Im r N , N . Throughout, the physical size N N N n n n
Near the planar double root, use E = E ∗ + s J 2 N − 4 / 5 X E=E^*+s_J^2N^{-4/5}X E = E ∗ + s J 2 N − 4/5 X t = 2 7 / 5 X + 4 1 / 5 ( n − N ) N − 1 / 5 t=2^{7/5}X+4^{1/5}(n-N)N^{-1/5} t = 2 7/5 X + 4 1/5 ( n − N ) N − 1/5
β n , N ≃ s J 2 + s J y ( t ) 2 1 / 5 N 2 / 5 , r n , N ≃ s J 2 4 − s J 2 y ( t ) 2 6 / 5 N 2 / 5 . (36) \beta_{n,N}\simeq\frac{s_J}{2}+\frac{s_Jy(t)}{2^{1/5}N^{2/5}},\qquad r_{n,N}\simeq\frac{s_J^2}{4}-\frac{s_J^2y(t)}{2^{6/5}N^{2/5}}. \tag*{(36)} β n , N ≃ 2 s J + 2 1/5 N 2/5 s J y ( t ) , r n , N ≃ 4 s J 2 − 2 6/5 N 2/5 s J 2 y ( t ) . (36) A unit shift of n n n t t t y ′ ′ = 6 y 2 + t y^{\prime\prime}=6y^2+t y ′′ = 6 y 2 + t n = N n=N n = N (30) .
The equation alone does not fix the exponentially small imaginary part responsible for the forbidden-side density; this is determined by the contour Γ − \Gamma_- Γ − λ = − E − ( J / 2 ) ( − E ) − 1 / 4 z \lambda=\sqrt{-E}-(J/2)(-E)^{-1/4}z λ = − E − ( J /2 ) ( − E ) − 1/4 z z 2 / 2 − u z 3 z^2/2-uz^3 z 2 /2 − u z 3 u = J / [ 12 ( − E ) 3 / 4 ] u=J/[12(-E)^{3/4}] u = J / [ 12 ( − E ) 3/4 ] E = E ∗ E=E^* E = E ∗ [57 , 58 ] , and the mapped outgoing contour is their α = 1 \alpha=1 α = 1 y 1 y_1 y 1
Its allowed-side behavior y 1 ∼ − i t / 6 y_1\sim-\mathrm{i}\sqrt{t/6} y 1 ∼ − i t /6 y 0 = − t / 6 y_0=\sqrt{-t/6} y 0 = − t /6 δ y ′ ′ = 12 y 0 δ y \delta y^{\prime\prime}=12y_0\delta y δ y ′′ = 12 y 0 δ y
S = ( 4 / 5 ) 2 1 / 4 ( − t ) 5 / 4 . S=(4/5)2^{1/4}(-t)^{5/4}. S = ( 4/5 ) 2 1/4 ( − t ) 5/4 . This yields Eq. (32) ; the contour fixes the amplitude and sign. The full double-scaling expansion and contour identification are given in [45 ] .
Model, conventions, Gelfand–Yaglom formula, and matrix oscillation
White-noise normalization and the matrix Riccati equation
We first make explicit the white-noise convention used in Eq. (1) of the Main Text. We write
H = − ∂ x 2 I N + W ( x ) , W ( x ) d x = J N d B ( x ) , (37) \mathcal{H}=-\partial_x^2 I_N+W(x),\qquad W(x)\mathrm{d}x=\frac{J}{\sqrt{N}}\mathrm{d}B(x), \tag*{(37)} H = − ∂ x 2 I N + W ( x ) , W ( x ) d x = N J d B ( x ) , (37) where B B B
E [ d B i j ( x ) d B k l ( x ) ] = δ i l δ j k d x . (38) \mathbb{E}[\mathrm{d}B_{ij}(x)\mathrm{d}B_{kl}(x)]=\delta_{il}\delta_{jk}\mathrm{d}x. \tag*{(38)} E [ d B ij ( x ) d B k l ( x )] = δ i l δ j k d x . (38) Equivalently, the Hermitian Brownian increment may be parametrized as
d B i i = d X i i , d B i j = d X i j + i d Y i j 2 , d B j i = d B i j ∗ = d X i j − i d Y i j 2 , i < j , (39) \mathrm{d}B_{ii}=\mathrm{d}X_{ii},\qquad\mathrm{d}B_{ij}=\frac{\mathrm{d}X_{ij}+i\mathrm{d}Y_{ij}}{\sqrt{2}},\qquad\mathrm{d}B_{ji}=\mathrm{d}B_{ij}^{*}=\frac{\mathrm{d}X_{ij}-i\mathrm{d}Y_{ij}}{\sqrt{2}},\qquad i<j, \tag*{(39)} d B ii = d X ii , d B ij = 2 d X ij + i d Y ij , d B j i = d B ij ∗ = 2 d X ij − i d Y ij , i < j , (39) where all d X i i \mathrm{d}X_{ii} d X ii d X i j \mathrm{d}X_{ij} d X ij d Y i j \mathrm{d}Y_{ij} d Y ij d x \mathrm{d}x d x B B B β = 2 \beta=2 β = 2 W d x = ( J / N ) d B W\mathrm{d}x=(J/\sqrt{N})\mathrm{d}B W d x = ( J / N ) d B
E [ W i j ( x ) W k l ( x ′ ) ] = J 2 N δ i l δ j k δ ( x − x ′ ) = 2 T δ i l δ j k δ ( x − x ′ ) , \mathbb{E}[W_{ij}(x)W_{kl}(x')]=\frac{J^2}{N}\delta_{il}\delta_{jk}\delta(x-x')=2T\delta_{il}\delta_{jk}\delta(x-x'), E [ W ij ( x ) W k l ( x ′ )] = N J 2 δ i l δ j k δ ( x − x ′ ) = 2 T δ i l δ j k δ ( x − x ′ ) , and hence
T = J 2 2 N , (40) T=\frac{J^2}{2N}, \tag*{(40)} T = 2 N J 2 , (40) as used throughout the Main Text. Let Y Y Y
Y ( 0 ) = 0 , Y ′ ( 0 ) = I N . Y(0)=0,\qquad Y'(0)=I_N. Y ( 0 ) = 0 , Y ′ ( 0 ) = I N . The corresponding first-order Hamiltonian system is formally
∂ x ( Y Y ′ ) = ( 0 I N W − E I N 0 ) ( Y Y ′ ) . (41) \partial_x\begin{pmatrix}Y\\Y'\end{pmatrix}=
\begin{pmatrix}0&I_N\\W-EI_N&0\end{pmatrix}
\begin{pmatrix}Y\\Y'\end{pmatrix}. \tag*{(41)} ∂ x ( Y Y ′ ) = ( 0 W − E I N I N 0 ) ( Y Y ′ ) . (41) For white noise, W W W (41) is understood through its integrated stochastic form. Whenever Y Y Y
Z : = Y ′ Y − 1 (42) Z:=Y'Y^{-1} \tag*{(42)} Z := Y ′ Y − 1 (42) satisfies
d Z = − ( E I N + Z 2 ) d x + J N d B . (43) \mathrm{d}Z=-(EI_N+Z^2)\mathrm{d}x+\frac{J}{\sqrt{N}}\mathrm{d}B. \tag*{(43)} d Z = − ( E I N + Z 2 ) d x + N J d B . (43) This is the stochastic form of Eq. (2) of the Main Text. Unitary invariance then gives the interacting eigenvalue diffusion Eq. (3) , i.e. the Dyson Brownian-motion reduction [42 ] . With the convention above, the scalar N = 1 N=1 N = 1 2 T δ ( x − x ′ ) 2T\delta(x-x') 2 T δ ( x − x ′ )
Gelfand–Yaglom and determinant growth
For finite L L L [31 , 32 , 33 ] give
det ( H L − E ) det ( − ∂ x 2 I N − E ) = det Y ( L ; E ) det Y 0 ( L ; E ) , (44) \frac{\det(\mathcal{H}_L-E)}{\det(-\partial_x^2I_N-E)}=\frac{\det Y(L;E)}{\det Y_0(L;E)}, \tag*{(44)} det ( − ∂ x 2 I N − E ) det ( H L − E ) = det Y 0 ( L ; E ) det Y ( L ; E ) , (44) where Y 0 Y_0 Y 0 Y 0 ( 0 ) = 0 Y_0(0)=0 Y 0 ( 0 ) = 0 Y 0 ′ ( 0 ) = I N Y_0'(0)=I_N Y 0 ′ ( 0 ) = I N
det Y ( L ; E ) = 0 ⟺ E ∈ Spec ( H L ) . (45) \det Y(L;E)=0 \Longleftrightarrow E\in\operatorname{Spec}(\mathcal{H}_L). \tag*{(45)} det Y ( L ; E ) = 0 ⟺ E ∈ Spec ( H L ) . (45) Thus singularities of Z = Y ′ Y − 1 Z=Y'Y^{-1} Z = Y ′ Y − 1 Y Y Y
∂ x log det Y ( x ) = Tr Z ( x ) = ∑ i = 1 N λ i ( x ) . (46) \partial_x\log\det Y(x)=\operatorname{Tr}Z(x)=\sum_{i=1}^{N}\lambda_i(x). \tag*{(46)} ∂ x log det Y ( x ) = Tr Z ( x ) = i = 1 ∑ N λ i ( x ) . (46) Hence the same Riccati process controls both spectral counting and longitudinal volume growth. In particular, Eq. (S10) is the identity used in the Main Text to express the total growth rate Γ N ( E ) = lim L → ∞ L − 1 log ∣ det Y ( L ) ∣ \Gamma_N(E)=\lim_{L\to\infty}L^{-1}\log|\det Y(L)| Γ N ( E ) = lim L → ∞ L − 1 log ∣ det Y ( L ) ∣
Matrix oscillation and the IDOS
For a smooth Hermitian matrix potential, the matrix Sturm oscillation theorem states that the number of Dirichlet eigenvalues of H L \mathcal{H}_L H L E E E Y ( x ; E ) Y(x;E) Y ( x ; E ) 0 < x ≤ L 0<x\le L 0 < x ≤ L [46 , 47 ] . A conjugate point is a point where det Y ( x ; E ) = 0 \det Y(x;E)=0 det Y ( x ; E ) = 0 Z = Y ′ Y − 1 Z=Y'Y^{-1} Z = Y ′ Y − 1
Let K L ( E ) K_L(E) K L ( E ) [ 0 , L ] [0,L] [ 0 , L ]
N L ( E ) = K L ( E ) + O ( 1 ) , (47) \mathcal{N}_L(E)=K_L(E)+O(1), \tag*{(47)} N L ( E ) = K L ( E ) + O ( 1 ) , (47) where N L ( E ) \mathcal{N}_L(E) N L ( E ) O ( 1 ) O(1) O ( 1 ) L L L K L ( E ) = N L ( E ) K_L(E)=\mathcal{N}_L(E) K L ( E ) = N L ( E )
ν N ( E ) = lim L → ∞ E N L ( E ) L = lim L → ∞ E K L ( E ) L . (48) \nu_N(E)=\lim_{L\to\infty}\frac{\mathbb{E}\mathcal{N}_L(E)}{L}=\lim_{L\to\infty}\frac{\mathbb{E}K_L(E)}{L}. \tag*{(48)} ν N ( E ) = L → ∞ lim L E N L ( E ) = L → ∞ lim L E K L ( E ) . (48) This establishes the spectral-counting–winding part of Eq. (5) of the Main Text. In Sec. II below we identify the limiting winding rate with the stationary flux through the projective boundary, thereby completing the current interpretation of Eq. (5) .
For N > 1 N>1 N > 1
Projective compactification and cyclic boundary condition
We spell out the global boundary condition underlying the Riccati dynamics and, in particular, its stationary determinantal solution. To make passages of the Riccati eigenvalues through infinity into ordinary phase windings, we introduce the Prüfer angle θ \theta θ κ = 1 \kappa=1 κ = 1
λ = cot θ , 0 < θ < π , θ ≡ θ + π . (49) \lambda=\cot\theta,\qquad0<\theta<\pi,\qquad\theta\equiv\theta+\pi. \tag*{(49)} λ = cot θ , 0 < θ < π , θ ≡ θ + π . (49) Thus λ = + ∞ \lambda=+\infty λ = + ∞ λ = − ∞ \lambda=-\infty λ = − ∞ θ = 0 \theta=0 θ = 0 θ = π \theta=\pi θ = π θ \theta θ π \pi π R P 1 \mathbb{R}\mathbb{P}^{1} R P 1
∣ d θ d λ ∣ = sin 2 θ = 1 1 + λ 2 , (50) \left|\frac{\mathrm{d}\theta}{\mathrm{d}\lambda}\right|=\sin^2\theta=\frac{1}{1+\lambda^2}, \tag*{(50)} d λ d θ = sin 2 θ = 1 + λ 2 1 , (50) so a density regular in θ \theta θ λ − 2 \lambda^{-2} λ − 2 λ i = cot θ i \lambda_i=\cot\theta_i λ i = cot θ i
d θ i = A i ( θ ) d x − 2 T sin 2 θ i d B i , A i ( θ ) = cos 2 θ i + E sin 2 θ i + 2 T sin 3 θ i ( cos θ i + ∑ j ≠ i sin θ j sin ( θ i − θ j ) ) , A i ⟶ 1 ( θ i → 0 , π ) . (51) \begin{aligned}
\mathrm{d}\theta_i &= A_i(\boldsymbol{\theta})\,\mathrm{d}x-\sqrt{2T}\sin^2\theta_i\,\mathrm{d}B_i,\\
A_i(\boldsymbol{\theta})&=\cos^2\theta_i+E\sin^2\theta_i+2T\sin^3\theta_i\left(\cos\theta_i+\sum_{j\ne i}\frac{\sin\theta_j}{\sin(\theta_i-\theta_j)}\right),\qquad A_i\longrightarrow1\quad(\theta_i\to0,\pi). \tag*{(51)}
\end{aligned} d θ i A i ( θ ) = A i ( θ ) d x − 2 T sin 2 θ i d B i , = cos 2 θ i + E sin 2 θ i + 2 T sin 3 θ i cos θ i + j = i ∑ sin ( θ i − θ j ) sin θ j , A i ⟶ 1 ( θ i → 0 , π ) . (51) If P ^ s t ( θ ) \widehat{P}_{\mathrm{st}}(\boldsymbol{\theta}) P st ( θ ) d N θ \mathrm{d}^N\theta d N θ ( θ 1 , … , θ N ) (\theta_1,\ldots,\theta_N) ( θ 1 , … , θ N ) θ i \theta_i θ i
J ^ i = A i P ^ s t − T ∂ θ i [ sin 4 θ i P ^ s t ] . (52) \widehat{\mathcal{J}}_i=A_i\widehat{P}_{\mathrm{st}}-T\partial_{\theta_i}\left[\sin^4\theta_i\widehat{P}_{\mathrm{st}}\right]. \tag*{(52)} J i = A i P st − T ∂ θ i [ sin 4 θ i P st ] . (52) Define the one-point phase density and the associated one-point current by
n ^ ( θ ) : = ⟨ ∑ i = 1 N δ ( θ − θ i ) ⟩ s t , j ( θ ) : = ∑ i = 1 N ∫ J ^ i δ ( θ − θ i ) ∏ k ≠ i d θ k . (53) \widehat{n}(\theta):=\left\langle\sum_{i=1}^{N}\delta(\theta-\theta_i)\right\rangle_{\mathrm{st}},\qquad j(\theta):=\sum_{i=1}^{N}\int\widehat{\mathcal{J}}_i\delta(\theta-\theta_i)\prod_{k\ne i}\mathrm{d}\theta_k. \tag*{(53)} n ( θ ) := ⟨ i = 1 ∑ N δ ( θ − θ i ) ⟩ st , j ( θ ) := i = 1 ∑ N ∫ J i δ ( θ − θ i ) k = i ∏ d θ k . (53) At the projective point θ = 0 ≡ π \theta=0\equiv\pi θ = 0 ≡ π sin 4 θ \sin^4\theta sin 4 θ A i → 1 A_i\to1 A i → 1
j ( 0 ) = n ^ ( 0 ) = j ( π ) = n ^ ( π ) = J s t w i n d , (54) j(0)=\widehat{n}(0)=j(\pi)=\widehat{n}(\pi)=\mathcal{J}_{\mathrm{st}}^{\mathrm{wind}}, \tag*{(54)} j ( 0 ) = n ( 0 ) = j ( π ) = n ( π ) = J st wind , (54) where J s t w i n d \mathcal{J}_{\mathrm{st}}^{\mathrm{wind}} J st wind (5) . The line and phase one-point densities are related by n ( λ ) ∣ d λ ∣ = n ^ ( θ ) d θ n(\lambda)|\mathrm{d}\lambda|=\widehat{n}(\theta)\mathrm{d}\theta n ( λ ) ∣ d λ ∣ = n ( θ ) d θ n ( λ ) = n ^ ( θ ) ∣ d θ / d λ ∣ n(\lambda)=\widehat{n}(\theta)|\mathrm{d}\theta/\mathrm{d}\lambda| n ( λ ) = n ( θ ) ∣ d θ / d λ ∣ (50) and J s t w i n d = ν N ( E ) \mathcal{J}_{\mathrm{st}}^{\mathrm{wind}}=\nu_N(E) J st wind = ν N ( E ) (5) , we therefore obtain
n ( λ ) ∼ J s t w i n d λ 2 = ν N ( E ) λ 2 , ∣ λ ∣ → ∞ . (55) n(\lambda)\sim\frac{\mathcal{J}_{\mathrm{st}}^{\mathrm{wind}}}{\lambda^2}=\frac{\nu_N(E)}{\lambda^2},\qquad|\lambda|\to\infty. \tag*{(55)} n ( λ ) ∼ λ 2 J st wind = λ 2 ν N ( E ) , ∣ λ ∣ → ∞. (55) Thus the universal λ − 2 \lambda^{-2} λ − 2
With the ordering convention of the main text, λ 1 < ⋯ < λ N \lambda_1<\cdots<\lambda_N λ 1 < ⋯ < λ N
0 < θ N < ⋯ < θ 1 < π . (56) 0<\theta_N<\cdots<\theta_1<\pi. \tag*{(56)} 0 < θ N < ⋯ < θ 1 < π . (56) When the smallest eigenvalue λ 1 \lambda_1 λ 1 − ∞ -\infty − ∞ θ 1 \theta_1 θ 1 π \pi π θ = 0 \theta=0 θ = 0 + ∞ +\infty + ∞
( λ 1 , λ 2 , … , λ N ) ⟶ ( λ 2 , … , λ N , + ∞ ) . (57) (\lambda_1,\lambda_2,\ldots,\lambda_N)\longrightarrow(\lambda_2,\ldots,\lambda_N,+\infty). \tag*{(57)} ( λ 1 , λ 2 , … , λ N ) ⟶ ( λ 2 , … , λ N , + ∞ ) . (57) This cyclic gluing, rather than separate boundary conditions at λ = ± ∞ \lambda=\pm\infty λ = ± ∞
The same geometry is visible in the Vandermonde. With V N ( λ ) = ∏ i < j ( λ j − λ i ) V_N(\lambda)=\prod_{i<j}(\lambda_j-\lambda_i) V N ( λ ) = ∏ i < j ( λ j − λ i )
V N ( λ ) = D N ( θ ) ∏ i = 1 N sin − ( N − 1 ) θ i , D N ( θ ) : = ∏ i < j sin ( θ i − θ j ) . (58) V_N(\lambda)=D_N(\boldsymbol{\theta})\prod_{i=1}^{N}\sin^{-(N-1)}\theta_i,\qquad D_N(\boldsymbol{\theta}):=\prod_{i<j}\sin(\theta_i-\theta_j). \tag*{(58)} V N ( λ ) = D N ( θ ) i = 1 ∏ N sin − ( N − 1 ) θ i , D N ( θ ) := i < j ∏ sin ( θ i − θ j ) . (58) At the projective boundary,
D N ( π , θ 2 , … , θ N ) = D N ( θ 2 , … , θ N , 0 ) , (59) D_N(\pi,\theta_2,\ldots,\theta_N)=D_N(\theta_2,\ldots,\theta_N,0), \tag*{(59)} D N ( π , θ 2 , … , θ N ) = D N ( θ 2 , … , θ N , 0 ) , (59) so the circular Vandermonde is continuous under reinjection followed by the cyclic relabelling (S21). The remaining factor in Eq. (58) matches as well: just before and after reinjection the escaping angle is respectively π − ε \pi-\varepsilon π − ε ε \varepsilon ε sin ( π − ε ) = sin ε \sin(\pi-\varepsilon)=\sin\varepsilon sin ( π − ε ) = sin ε
We now show that the biorthogonal stationary solution Eq. (14) obeys this gluing. Every polynomial eigenmode p a p_a p a N − 1 N-1 N − 1 ℓ N † \ell_N^\dagger ℓ N † (10) . The unit-concomitant condition Eq. (15) then implies v a ( λ ) = O ( λ − N − 1 ) v_a(\lambda)=O(\lambda^{-N-1}) v a ( λ ) = O ( λ − N − 1 ) λ → ± ∞ \lambda\to\pm\infty λ → ± ∞
p ^ a ( θ ) = sin N − 1 θ p a ( cot θ ) , v ^ a ( θ ) = sin − N − 1 θ v a ( cot θ ) . (60) \hat{p}_a(\theta)=\sin^{N-1}\theta\,p_a(\cot\theta),\qquad\hat{v}_a(\theta)=\sin^{-N-1}\theta\,v_a(\cot\theta). \tag*{(60)} p ^ a ( θ ) = sin N − 1 θ p a ( cot θ ) , v ^ a ( θ ) = sin − N − 1 θ v a ( cot θ ) . (60) Both are regular at the projective point. Since cot ( θ + π ) = cot θ \cot(\theta+\pi)=\cot\theta cot ( θ + π ) = cot θ sin ( θ + π ) = − sin θ \sin(\theta+\pi)=-\sin\theta sin ( θ + π ) = − sin θ
p ^ a ( θ + π ) = ( − 1 ) N − 1 p ^ a ( θ ) , v ^ a ( θ + π ) = ( − 1 ) N − 1 v ^ a ( θ ) . (61) \hat{p}_a(\theta+\pi)=(-1)^{N-1}\hat{p}_a(\theta),\qquad\hat{v}_a(\theta+\pi)=(-1)^{N-1}\hat{v}_a(\theta). \tag*{(61)} p ^ a ( θ + π ) = ( − 1 ) N − 1 p ^ a ( θ ) , v ^ a ( θ + π ) = ( − 1 ) N − 1 v ^ a ( θ ) . (61) The compactification is also natural directly at the level of the stationary measure. Using ∣ d λ ∣ = d θ / sin 2 θ |\mathrm{d}\lambda|=\mathrm{d}\theta/\sin^2\theta ∣ d λ ∣ = d θ / sin 2 θ (14) , all powers of sin θ i \sin\theta_i sin θ i
P s t ( λ ) d N λ ∝ det a , i [ p ^ a ( θ i ) ] det a , i [ v ^ a ( θ i ) ] d N θ . (62) P_{\mathrm{st}}(\lambda)\,\mathrm{d}^N\lambda\propto\det_{a,i}[\hat{p}_a(\theta_i)]\det_{a,i}[\hat{v}_a(\theta_i)]\,\mathrm{d}^N\theta. \tag*{(62)} P st ( λ ) d N λ ∝ a , i det [ p ^ a ( θ i )] a , i det [ v ^ a ( θ i )] d N θ . (62) Under one reinjection, restoring the ordered labels cyclically permutes the columns of either Slater determinant, ( 1 , 2 , … , N ) ↦ ( 2 , … , N , 1 ) (1,2,\ldots,N)\mapsto(2,\ldots,N,1) ( 1 , 2 , … , N ) ↦ ( 2 , … , N , 1 ) ( − 1 ) N − 1 (-1)^{N-1} ( − 1 ) N − 1 ( − 1 ) N − 1 (-1)^{N-1} ( − 1 ) N − 1 (61) . The two factors cancel, so each Slater determinant, and hence the stationary measure and current, is single-valued under cyclic reinjection.
This also identifies the global boundary condition selecting the physical right mode v a v_a v a ℓ N v a = ε a v a \ell_Nv_a=\varepsilon_av_a ℓ N v a = ε a v a v ^ a \hat{v}_a v ^ a (61) . Expressed in the λ \lambda λ (16) . Consequently, the free-fermion representation derived in Eq. (8) is not only valid locally in the bulk: its Slater determinants also satisfy the cyclic reinjection condition at infinity.
For comparison with the matrix Prüfer formulation, restore an arbitrary scale κ > 0 \kappa>0 κ > 0 λ i = κ cot θ i \lambda_i=\kappa\cot\theta_i λ i = κ cot θ i
U P ( x ) = [ Z ( x ) + i κ I N ] [ Z ( x ) − i κ I N ] − 1 (63) U_{\mathrm{P}}(x)=[Z(x)+\mathrm{i}\kappa I_N][Z(x)-\mathrm{i}\kappa I_N]^{-1} \tag*{(63)} U P ( x ) = [ Z ( x ) + i κ I N ] [ Z ( x ) − i κ I N ] − 1 (63) is unitary because Z Z Z λ i \lambda_i λ i Z Z Z U P U_{\mathrm{P}} U P e 2 i θ i \mathrm{e}^{2\mathrm{i}\theta_i} e 2 i θ i U P U_{\mathrm{P}} U P e 2 i θ i \mathrm{e}^{2\mathrm{i}\theta_i} e 2 i θ i
The same Cayley transform has a standard scattering interpretation [72 , 73 ] . To see it, consider as an auxiliary construction the disordered interval [ 0 , L ] [0,L] [ 0 , L ] x = 0 x=0 x = 0 x = L x=L x = L N N N κ \kappa κ a \boldsymbol{a} a b \boldsymbol{b} b b = S L a \boldsymbol{b}=S_L\boldsymbol{a} b = S L a S L S_L S L ψ ( x ) = e − i κ ( x − L ) a + e i κ ( x − L ) b \psi(x)=\mathrm{e}^{-\mathrm{i}\kappa(x-L)}\boldsymbol{a}+\mathrm{e}^{\mathrm{i}\kappa(x-L)}\boldsymbol{b} ψ ( x ) = e − i κ ( x − L ) a + e i κ ( x − L ) b x = L x=L x = L Z L : = Z ( L ; E ) Z_L:=Z(L;E) Z L := Z ( L ; E )
S L = − [ Z L − i κ I N ] − 1 [ Z L + i κ I N ] = − U P ( L ) . (64) S_L=-[Z_L-\mathrm{i}\kappa I_N]^{-1}[Z_L+\mathrm{i}\kappa I_N]=-U_{\mathrm{P}}(L). \tag*{(64)} S L = − [ Z L − i κ I N ] − 1 [ Z L + i κ I N ] = − U P ( L ) . (64) Thus a Riccati winding can equivalently be viewed as a winding of a reflection eigenvalue around the unit circle. This scattering interpretation is auxiliary; the projective compactification itself does not require attaching leads.
The same projective boundary condition also provides a natural way to count Riccati windings: assigning a factor z z z
Exact β = 2 \beta=2 β = 2
We give the details of the cancellation leading to Eq. (8) . In the ordered chamber C N \mathcal{C}_N C N (4) is
L N F P P = ∑ i = 1 N { T ∂ i 2 P + ∂ i [ ( E + λ i 2 ) P ] − 2 T ∂ i [ P ∑ j ≠ i 1 λ i − λ j ] } . (65) \mathcal{L}^{\mathrm{FP}}_N P=\sum_{i=1}^{N}\left\{T\partial_i^2P+\partial_i\left[(E+\lambda_i^2)P\right]-2T\partial_i\left[P\sum_{j\ne i}\frac{1}{\lambda_i-\lambda_j}\right]\right\}. \tag*{(65)} L N FP P = i = 1 ∑ N ⎩ ⎨ ⎧ T ∂ i 2 P + ∂ i [ ( E + λ i 2 ) P ] − 2 T ∂ i P j = i ∑ λ i − λ j 1 ⎭ ⎬ ⎫ . (65) Set P = V N Ψ P=V_N\Psi P = V N Ψ V N ( λ ) = ∏ i < j ( λ j − λ i ) V_N(\lambda)=\prod_{i<j}(\lambda_j-\lambda_i) V N ( λ ) = ∏ i < j ( λ j − λ i ) λ 1 < ⋯ < λ N \lambda_1<\cdots<\lambda_N λ 1 < ⋯ < λ N V N > 0 V_N>0 V N > 0 C N C_N C N P P P P P P P P P V N V_N V N Ψ = P / V N \Psi=P/V_N Ψ = P / V N
∂ i V N V N = ∑ j ≠ i 1 λ i − λ j , ∑ i = 1 N ∂ i 2 V N = 0 , (66) \frac{\partial_i V_N}{V_N}=\sum_{j\ne i}\frac{1}{\lambda_i-\lambda_j},\qquad\sum_{i=1}^{N}\partial_i^2V_N=0, \tag*{(66)} V N ∂ i V N = j = i ∑ λ i − λ j 1 , i = 1 ∑ N ∂ i 2 V N = 0 , (66) produce the usual β = 2 \beta=2 β = 2 U E ′ ( λ ) = E + λ 2 U_E'(\lambda)=E+\lambda^2 U E ′ ( λ ) = E + λ 2
∑ i = 1 N U E ′ ( λ i ) ∑ j ≠ i 1 λ i − λ j = ∑ i < j U E ′ ( λ i ) − U E ′ ( λ j ) λ i − λ j = ( N − 1 ) ∑ i = 1 N λ i . (67) \sum_{i=1}^{N}U_E'(\lambda_i)\sum_{j\ne i}\frac{1}{\lambda_i-\lambda_j}
=\sum_{i<j}\frac{U_E'(\lambda_i)-U_E'(\lambda_j)}{\lambda_i-\lambda_j}
=(N-1)\sum_{i=1}^{N}\lambda_i. \tag*{(67)} i = 1 ∑ N U E ′ ( λ i ) j = i ∑ λ i − λ j 1 = i < j ∑ λ i − λ j U E ′ ( λ i ) − U E ′ ( λ j ) = ( N − 1 ) i = 1 ∑ N λ i . (67) Together with U E ′ ′ ( λ ) = 2 λ U_E''(\lambda)=2\lambda U E ′′ ( λ ) = 2 λ ( N + 1 ) λ (N+1)\lambda ( N + 1 ) λ
L N F P [ V N Ψ ] = V N ∑ i = 1 N ℓ N ( i ) Ψ , ℓ N = T ∂ λ 2 + ( E + λ 2 ) ∂ λ + ( N + 1 ) λ , (68) \mathcal{L}^{\mathrm{FP}}_N[V_N\Psi]=V_N\sum_{i=1}^{N}\ell_N^{(i)}\Psi,\qquad
\ell_N=T\partial_\lambda^2+(E+\lambda^2)\partial_\lambda+(N+1)\lambda, \tag*{(68)} L N FP [ V N Ψ ] = V N i = 1 ∑ N ℓ N ( i ) Ψ , ℓ N = T ∂ λ 2 + ( E + λ 2 ) ∂ λ + ( N + 1 ) λ , (68) with adjoint
ℓ N † = T ∂ λ 2 − ( E + λ 2 ) ∂ λ + ( N − 1 ) λ . (69) \ell_N^\dagger=T\partial_\lambda^2-(E+\lambda^2)\partial_\lambda+(N-1)\lambda. \tag*{(69)} ℓ N † = T ∂ λ 2 − ( E + λ 2 ) ∂ λ + ( N − 1 ) λ . (69) Thus the transformed bulk generator is exactly a sum of one-body operators at every finite N N N (66) is the familiar unitary-class fermionization, whereas Eq. (67) , which uses the quadratic Riccati drift, is the additional special property of the present model. The compatibility of this bulk reduction with the cyclic projective boundary condition, including the compensating fermionic sign under reinjection, was established in the preceding section.
Polynomial sector and biorthogonal stationary process
Finite-dimensional polynomial sector and electrostatic roots
The adjoint operator (S33) preserves P N − 1 = span { 1 , λ , … , λ N − 1 } \mathcal{P}_{N-1}=\operatorname{span}\{1,\lambda,\ldots,\lambda^{N-1}\} P N − 1 = span { 1 , λ , … , λ N − 1 }
ℓ N † λ n = T n ( n − 1 ) λ n − 2 − E n λ n − 1 + ( N − 1 − n ) λ n + 1 , 0 ≤ n ≤ N − 1. (70) \ell_N^\dagger\lambda^n=Tn(n-1)\lambda^{n-2}-En\lambda^{n-1}+(N-1-n)\lambda^{n+1},\qquad0\le n\le N-1. \tag*{(70)} ℓ N † λ n = T n ( n − 1 ) λ n − 2 − E n λ n − 1 + ( N − 1 − n ) λ n + 1 , 0 ≤ n ≤ N − 1. (70) In the monomial basis ( 1 , λ , … , λ N − 1 ) (1,\lambda,\ldots,\lambda^{N-1}) ( 1 , λ , … , λ N − 1 ) M N M_N M N ℓ N † ∣ P N − 1 \ell_N^\dagger|_{\mathcal{P}_{N-1}} ℓ N † ∣ P N − 1 m , n = 0 , … , N − 1 m,n=0,\ldots,N-1 m , n = 0 , … , N − 1
( M N ) m n = T n ( n − 1 ) δ m , n − 2 − E n δ m , n − 1 + ( N − 1 − n ) δ m , n + 1 . (71) (M_N)_{mn}=Tn(n-1)\delta_{m,n-2}-En\delta_{m,n-1}+(N-1-n)\delta_{m,n+1}. \tag*{(71)} ( M N ) mn = T n ( n − 1 ) δ m , n − 2 − E n δ m , n − 1 + ( N − 1 − n ) δ m , n + 1 . (71) Thus the finite-N N N M N M_N M N
Q N ( ε ) : = det ( ε I N − M N ) , (72) Q_N(\varepsilon):=\det(\varepsilon I_N-M_N), \tag*{(72)} Q N ( ε ) := det ( ε I N − M N ) , (72) the first cases are
Q 1 ( ε ) = ε , Q 2 ( ε ) = ε 2 + E , Q 3 ( ε ) = ε 3 + 4 E ε − 4 T , Q 4 ( ε ) = ε 4 + 10 E ε 2 − 24 T ε + 9 E 2 . (73) \begin{aligned}
Q_1(\varepsilon)&=\varepsilon,\\
Q_2(\varepsilon)&=\varepsilon^2+E,\\
Q_3(\varepsilon)&=\varepsilon^3+4E\varepsilon-4T,\\
Q_4(\varepsilon)&=\varepsilon^4+10E\varepsilon^2-24T\varepsilon+9E^2. \tag*{(73)}
\end{aligned} Q 1 ( ε ) Q 2 ( ε ) Q 3 ( ε ) Q 4 ( ε ) = ε , = ε 2 + E , = ε 3 + 4 E ε − 4 T , = ε 4 + 10 E ε 2 − 24 T ε + 9 E 2 . (73) A useful feature is visible directly in Eq. (70) : there is no term proportional to λ n \lambda^n λ n M N M_N M N
∑ a = 1 N ε a = Tr M N = 0 , (74) \sum_{a=1}^{N}\varepsilon_a=\operatorname{Tr}M_N=0, \tag*{(74)} a = 1 ∑ N ε a = Tr M N = 0 , (74) where the eigenvalues are counted with algebraic multiplicity. For generic parameters M N M_N M N N N N p a p_a p a (11) . Equation (74) is precisely what makes the filled N N N (13) .
Using the monic normalization of the polynomial modes, with the corresponding physical right partners subsequently normalized by B [ p a , v a ] = 1 B[p_a,v_a]=1 B [ p a , v a ] = 1
N = 1 : ε 1 = 0 , p 1 ( λ ) = 1 , N = 2 : ε a 2 + E = 0 , p a ( λ ) = λ + ε a , a = 1 , 2 , N = 3 : ε a 3 + 4 E ε a − 4 T = 0 , p a ( λ ) = λ 2 + ε a λ + ε a 2 + 2 E 2 , a = 1 , 2 , 3. (75) \begin{aligned}
N=1 &: \quad\varepsilon_1=0,\qquad p_1(\lambda)=1,\\
N=2 &: \quad\varepsilon_a^2+E=0,\qquad p_a(\lambda)=\lambda+\varepsilon_a,\qquad a=1,2,\\
N=3 &: \quad\varepsilon_a^3+4E\varepsilon_a-4T=0,\qquad p_a(\lambda)=\lambda^2+\varepsilon_a\lambda+\frac{\varepsilon_a^2+2E}{2},\qquad a=1,2,3. \tag*{(75)}
\end{aligned} N = 1 N = 2 N = 3 : ε 1 = 0 , p 1 ( λ ) = 1 , : ε a 2 + E = 0 , p a ( λ ) = λ + ε a , a = 1 , 2 , : ε a 3 + 4 E ε a − 4 T = 0 , p a ( λ ) = λ 2 + ε a λ + 2 ε a 2 + 2 E , a = 1 , 2 , 3. (75) Thus, at fixed N N N N N N p a p_a p a N N N N − 1 N-1 N − 1 P N − 1 \mathcal{P}_{N-1} P N − 1
Every nonzero polynomial eigenmode has degree exactly N − 1 N-1 N − 1 p ( λ ) = c m λ m + ⋯ p(\lambda)=c_m\lambda^m+\cdots p ( λ ) = c m λ m + ⋯ m < N − 1 m<N-1 m < N − 1 (70) produces a nonzero contribution of degree m + 1 m+1 m + 1 ℓ N † p = ε p \ell_N^\dagger p=\varepsilon p ℓ N † p = εp
p a ( λ ) = ∏ j = 1 N − 1 ( λ − r j ( a ) ) . (76) p_a(\lambda)=\prod_{j=1}^{N-1}\left(\lambda-r_j^{(a)}\right). \tag*{(76)} p a ( λ ) = j = 1 ∏ N − 1 ( λ − r j ( a ) ) . (76) At a simple zero r j ( a ) r_j^{(a)} r j ( a ) T p a ′ ′ − ( E + λ 2 ) p a ′ = 0 T p_a^{\prime\prime}-(E+\lambda^2)p_a^\prime=0 T p a ′′ − ( E + λ 2 ) p a ′ = 0 p a ′ ′ ( r j ) / p a ′ ( r j ) = 2 ∑ k ≠ j ( r j − r k ) − 1 p_a^{\prime\prime}(r_j)/p_a^\prime(r_j)=2\sum_{k\ne j}(r_j-r_k)^{-1} p a ′′ ( r j ) / p a ′ ( r j ) = 2 ∑ k = j ( r j − r k ) − 1
E + ( r j ( a ) ) 2 = 2 T ∑ k ≠ j 1 r j ( a ) − r k ( a ) , j = 1 , … , N − 1. (77) E+\left(r_j^{(a)}\right)^2=2T\sum_{k\ne j}\frac{1}{r_j^{(a)}-r_k^{(a)}},\qquad j=1,\ldots,N-1. \tag*{(77)} E + ( r j ( a ) ) 2 = 2 T k = j ∑ r j ( a ) − r k ( a ) 1 , j = 1 , … , N − 1. (77) The roots r j ( a ) r_j^{(a)} r j ( a ) N N N p a p_a p a N N N { r 1 ( a ) , … , r N − 1 ( a ) } \{r_1^{(a)},\ldots,r_{N-1}^{(a)}\} { r 1 ( a ) , … , r N − 1 ( a ) } (77) ; the label a a a N = 2 N=2 N = 2 E + r 2 = 0 E+r^2=0 E + r 2 = 0 E < 0 E<0 E < 0 r = ± − E r=\pm\sqrt{-E} r = ± − E E > 0 E>0 E > 0 r = ± i E r=\pm i\sqrt{E} r = ± i E λ N − 1 \lambda^{N-1} λ N − 1 ℓ N † p a = ε a p a \ell_N^\dagger p_a=\varepsilon_a p_a ℓ N † p a = ε a p a
ε a = − ∑ j = 1 N − 1 r j ( a ) = − ( N − 1 ) r ˉ a . (78) \varepsilon_a=-\sum_{j=1}^{N-1}r_j^{(a)}=-(N-1)\bar r_a. \tag*{(78)} ε a = − j = 1 ∑ N − 1 r j ( a ) = − ( N − 1 ) r ˉ a . (78) Equations (S41) are the stationary-point equations of the finite-dimensional logarithmic energy
W E ( r ) = ∑ j = 1 N − 1 U E ( r j ) − 2 T ∑ j < k log ( r j − r k ) , U E ( r ) = E r + r 3 3 . (79) \mathcal{W}_E(\mathbf r)=\sum_{j=1}^{N-1}U_E(r_j)-2T\sum_{j<k}\log(r_j-r_k),\qquad U_E(r)=Er+\frac{r^3}{3}. \tag*{(79)} W E ( r ) = j = 1 ∑ N − 1 U E ( r j ) − 2 T j < k ∑ log ( r j − r k ) , U E ( r ) = E r + 3 r 3 . (79) For real r i r_i r i log ( r j − r k ) \log(r_j-r_k) log ( r j − r k ) log ∣ r j − r k ∣ \log\lvert r_j-r_k\rvert log ∣ r j − r k ∣ r j ( a ) r_j^{(a)} r j ( a ) (79) is then understood by analytic continuation. There is consequently no ordering of such complex configurations by a real “electrostatic energy.” Moreover, even on the real axis the cubic external potential is nonconfining, so the solutions are stationary configurations rather than global minima.
For each exact polynomial mode p a p_a p a { r 1 ( a ) , … , r N − 1 ( a ) } \{r_1^{(a)},\ldots,r_{N-1}^{(a)}\} { r 1 ( a ) , … , r N − 1 ( a ) } N N N a a a § 0.9 , for any sequence of modes whose roots condense onto a smooth limiting contour, the finite-N N N [34 ] . Different sequences may correspond to different solutions or analytic branches of this planar equation; the physical Dyson–Riccati problem selects the appropriate large-N N N N N N
Physical right partners and a regular construction
For each polynomial mode p a p_a p a ℓ N v a = ε a v a \ell_N v_a=\varepsilon_a v_a ℓ N v a = ε a v a (15) ,
T ( p a v a ′ − p a ′ v a ) + ( E + λ 2 ) p a v a = 1. (80) T(p_a v_a' - p_a'v_a) + (E+\lambda^2)p_a v_a = 1. \tag*{(80)} T ( p a v a ′ − p a ′ v a ) + ( E + λ 2 ) p a v a = 1. (80) Since p a p_a p a N − 1 N-1 N − 1
p a ( λ ) v a ( λ ) ∼ 1 E + λ 2 , v a ( λ ) = O ( λ − N − 1 ) , λ → ± ∞ , (81) p_a(\lambda)v_a(\lambda) \sim\frac{1}{E+\lambda^2}, \qquad v_a(\lambda)=O(\lambda^{-N-1}), \qquad\lambda\to\pm\infty, \tag*{(81)} p a ( λ ) v a ( λ ) ∼ E + λ 2 1 , v a ( λ ) = O ( λ − N − 1 ) , λ → ± ∞ , (81) in agreement with the compactified regularity derived in the preceding section. Dividing Eq. (80) by T p a 2 Tp_a^2 T p a 2
d d λ [ e U E ( λ ) / T v a ( λ ) p a ( λ ) ] = 1 T e U E ( λ ) / T p a ( λ ) 2 . (82) \frac{\mathrm{d}}{\mathrm{d}\lambda}\left[\frac{e^{U_E(\lambda)/T}v_a(\lambda)}{p_a(\lambda)}\right]=\frac{1}{T}\frac{e^{U_E(\lambda)/T}}{p_a(\lambda)^2}. \tag*{(82)} d λ d [ p a ( λ ) e U E ( λ ) / T v a ( λ ) ] = T 1 p a ( λ ) 2 e U E ( λ ) / T . (82) Before integrating, note that the lower endpoint causes no ambiguity. As y → − ∞ y\to-\infty y → − ∞ U E ( y ) ∼ y 3 / 3 → − ∞ U_E(y)\sim y^3/3\to-\infty U E ( y ) ∼ y 3 /3 → − ∞ p a ( y ) ∼ y N − 1 p_a(y)\sim y^{N-1} p a ( y ) ∼ y N − 1 e U E ( y ) / T / p a ( y ) 2 e^{U_E(y)/T}/p_a(y)^2 e U E ( y ) / T / p a ( y ) 2 − ∞ -\infty − ∞ (81) ,
e U E ( y ) / T v a ( y ) p a ( y ) ⟶ 0 , y → − ∞ . \frac{e^{U_E(y)/T}v_a(y)}{p_a(y)}\longrightarrow0,\qquad y\to-\infty. p a ( y ) e U E ( y ) / T v a ( y ) ⟶ 0 , y → − ∞. We may therefore integrate Eq. (82) from − ∞ -\infty − ∞
v a ( λ ) = 1 T e − U E ( λ ) / T p a ( λ ) FP ∫ − ∞ λ e U E ( y ) / T p a ( y ) 2 d y . (83) v_a(\lambda)=\frac{1}{T}e^{-U_E(\lambda)/T}p_a(\lambda)\operatorname{FP}\int_{-\infty}^{\lambda}\frac{e^{U_E(y)/T}}{p_a(y)^2}\,\mathrm{d}y. \tag*{(83)} v a ( λ ) = T 1 e − U E ( λ ) / T p a ( λ ) FP ∫ − ∞ λ p a ( y ) 2 e U E ( y ) / T d y . (83) Here FP denotes the Hadamard finite part; it is needed only when p a p_a p a v a v_a v a r r r p a p_a p a y = r y=r y = r
e U E ( y ) / T p a ( y ) 2 = e U E ( r ) / T p a ′ ( r ) 2 [ 1 ( y − r ) 2 + ( U E ′ ( r ) T − p a ′ ′ ( r ) p a ′ ( r ) ) 1 y − r + O ( 1 ) ] . (84) \frac{e^{U_E(y)/T}}{p_a(y)^2}=\frac{e^{U_E(r)/T}}{p_a'(r)^2}\left[\frac{1}{(y-r)^2}+\left(\frac{U_E'(r)}{T}-\frac{p_a''(r)}{p_a'(r)}\right)\frac{1}{y-r}+O(1)\right]. \tag*{(84)} p a ( y ) 2 e U E ( y ) / T = p a ′ ( r ) 2 e U E ( r ) / T [ ( y − r ) 2 1 + ( T U E ′ ( r ) − p a ′ ( r ) p a ′′ ( r ) ) y − r 1 + O ( 1 ) ] . (84) At a zero of p a p_a p a ℓ N † p a = ε a p a \ell_N^\dagger p_a=\varepsilon_a p_a ℓ N † p a = ε a p a T p a ′ ′ ( r ) = U E ′ ( r ) p a ′ ( r ) = ( E + r 2 ) p a ′ ( r ) Tp_a''(r)=U_E'(r)p_a'(r)=(E+r^2)p_a'(r) T p a ′′ ( r ) = U E ′ ( r ) p a ′ ( r ) = ( E + r 2 ) p a ′ ( r ) (77) . The coefficient of ( y − r ) − 1 (y-r)^{-1} ( y − r ) − 1
e U E ( y ) / T p a ( y ) 2 = e U E ( r ) / T p a ′ ( r ) 2 [ 1 ( y − r ) 2 + O ( 1 ) ] , \frac{e^{U_E(y)/T}}{p_a(y)^2}=\frac{e^{U_E(r)/T}}{p_a'(r)^2}\left[\frac{1}{(y-r)^2}+O(1)\right], p a ( y ) 2 e U E ( y ) / T = p a ′ ( r ) 2 e U E ( r ) / T [ ( y − r ) 2 1 + O ( 1 ) ] , and its finite-part primitive behaves as
− e U E ( r ) / T p a ′ ( r ) 2 1 λ − r + O ( 1 ) ( λ → r ) , -\frac{e^{U_E(r)/T}}{p_a'(r)^2}\frac{1}{\lambda-r}+O(1)\qquad(\lambda\to r), − p a ′ ( r ) 2 e U E ( r ) / T λ − r 1 + O ( 1 ) ( λ → r ) , with no logarithmic term. The simple pole of the primitive is precisely cancelled by p a ( λ ) = p a ′ ( r ) ( λ − r ) + ⋯ p_a(\lambda)=p_a'(r)(\lambda-r)+\cdots p a ( λ ) = p a ′ ( r ) ( λ − r ) + ⋯ (83) . Hence v a v_a v a r r r
v a ( r ) = − 1 T p a ′ ( r ) . (85) v_a(r)=-\frac{1}{Tp_a'(r)}. \tag*{(85)} v a ( r ) = − T p a ′ ( r ) 1 . (85) Let us give the first two cases explicitly. For N = 1 N=1 N = 1 p 1 = 1 p_1=1 p 1 = 1 ε 1 = 0 \varepsilon_1=0 ε 1 = 0
v 1 ( λ ) = 1 T e − U E ( λ ) / T ∫ − ∞ λ e U E ( y ) / T d y , (86) v_1(\lambda)=\frac{1}{T}e^{-U_E(\lambda)/T}\int_{-\infty}^{\lambda}e^{U_E(y)/T}\,dy, \tag*{(86)} v 1 ( λ ) = T 1 e − U E ( λ ) / T ∫ − ∞ λ e U E ( y ) / T d y , (86) which is Halperin’s constant-current solution. For N = 2 N=2 N = 2 ε a 2 + E = 0 \varepsilon_a^2+E=0 ε a 2 + E = 0 p a ( λ ) = λ + ε a p_a(\lambda)=\lambda+\varepsilon_a p a ( λ ) = λ + ε a (83) gives
v a ( λ ) = 1 T e − U E ( λ ) / T ( λ + ε a ) FP ∫ − ∞ λ e U E ( y ) / T ( y + ε a ) 2 d y . (87) v_a(\lambda)=\frac{1}{T}e^{-U_E(\lambda)/T}(\lambda+\varepsilon_a)\operatorname{FP}\int_{-\infty}^{\lambda}\frac{e^{U_E(y)/T}}{(y+\varepsilon_a)^2}\,dy. \tag*{(87)} v a ( λ ) = T 1 e − U E ( λ ) / T ( λ + ε a ) FP ∫ − ∞ λ ( y + ε a ) 2 e U E ( y ) / T d y . (87) Using E = − ε a 2 E=-\varepsilon_a^2 E = − ε a 2
e U E ( y ) / T ( y + ε a ) 2 = y − ε a T e U E ( y ) / T − d d y [ e U E ( y ) / T y + ε a ] , \frac{e^{U_E(y)/T}}{(y+\varepsilon_a)^2}=\frac{y-\varepsilon_a}{T}e^{U_E(y)/T}-\frac{\mathrm{d}}{\mathrm{d}y}\left[\frac{e^{U_E(y)/T}}{y+\varepsilon_a}\right], ( y + ε a ) 2 e U E ( y ) / T = T y − ε a e U E ( y ) / T − d y d [ y + ε a e U E ( y ) / T ] , and hence the same regular solution can be written without a finite part as
v a ( λ ) = − 1 T + λ + ε a T 2 e − U E ( λ ) / T ∫ − ∞ λ ( y − ε a ) e U E ( y ) / T d y . (88) v_a(\lambda)=-\frac{1}{T}+\frac{\lambda+\varepsilon_a}{T^2}e^{-U_E(\lambda)/T}\int_{-\infty}^{\lambda}(y-\varepsilon_a)e^{U_E(y)/T}\,dy. \tag*{(88)} v a ( λ ) = − T 1 + T 2 λ + ε a e − U E ( λ ) / T ∫ − ∞ λ ( y − ε a ) e U E ( y ) / T d y . (88) When ε a \varepsilon_a ε a y = − ε a y=-\varepsilon_a y = − ε a v a ( − ε a ) = − 1 / T v_a(-\varepsilon_a)=-1/T v a ( − ε a ) = − 1/ T N N N N N N
For analytic work Eq. (83) is often the shortest representation. For numerical work, however, it is preferable to avoid finite parts altogether and integrate directly the first-order concomitant equation
T p a v a ′ + [ ( E + λ 2 ) p a − T p a ′ ] v a = 1 , (89) Tp_av_a'+\left[(E+\lambda^2)p_a-Tp_a'\right]v_a=1, \tag*{(89)} T p a v a ′ + [ ( E + λ 2 ) p a − T p a ′ ] v a = 1 , (89) using Eq. (85) to continue the regular solution smoothly across each real zero of p a p_a p a
Biorthogonality, stationary kernel, and the IDOS
We now use the left/right structure of the one-particle problem to construct the stationary determinantal process. The polynomial modes p a p_a p a v a v_a v a ℓ N † p a = ε a p a \ell_N^\dagger p_a=\varepsilon_a p_a ℓ N † p a = ε a p a ℓ N v a = ε a v a \ell_Nv_a=\varepsilon_av_a ℓ N v a = ε a v a
⟨ p a , v b ⟩ : = ∫ − ∞ ∞ p a ( λ ) v b ( λ ) d λ . (90) \langle p_a,v_b\rangle:=\int_{-\infty}^{\infty}p_a(\lambda)v_b(\lambda)\,\mathrm{d}\lambda. \tag*{(90)} ⟨ p a , v b ⟩ := ∫ − ∞ ∞ p a ( λ ) v b ( λ ) d λ . (90) No complex conjugation is involved: p a p_a p a v b v_b v b p a ( λ ) v b ( λ ) = O ( λ − 2 ) p_a(\lambda)v_b(\lambda)=O(\lambda^{-2}) p a ( λ ) v b ( λ ) = O ( λ − 2 )
For arbitrary functions p p p v v v
B [ p , v ] ( λ ) : = T ( p ( λ ) v ′ ( λ ) − p ′ ( λ ) v ( λ ) ) + ( E + λ 2 ) p ( λ ) v ( λ ) . (91) \mathcal{B}[p,v](\lambda):=T\left(p(\lambda)v'(\lambda)-p'(\lambda)v(\lambda)\right)+(E+\lambda^2)p(\lambda)v(\lambda). \tag*{(91)} B [ p , v ] ( λ ) := T ( p ( λ ) v ′ ( λ ) − p ′ ( λ ) v ( λ ) ) + ( E + λ 2 ) p ( λ ) v ( λ ) . (91) This is the boundary bilinear form associated with the adjoint pair ℓ N † , ℓ N \ell_N^\dagger,\ell_N ℓ N † , ℓ N
p ℓ N v − v ℓ N † p = ∂ λ B [ p , v ] . (92) p\ell_Nv-v\ell_N^\dagger p=\partial_\lambda\mathcal{B}[p,v]. \tag*{(92)} p ℓ N v − v ℓ N † p = ∂ λ B [ p , v ] . (92) For a pair of eigenmodes this becomes
∂ λ B [ p a , v b ] = ( ε b − ε a ) p a v b . (93) \partial_\lambda\mathcal{B}[p_a,v_b]=(\varepsilon_b-\varepsilon_a)p_av_b. \tag*{(93)} ∂ λ B [ p a , v b ] = ( ε b − ε a ) p a v b . (93) Integrating along the Riccati line therefore gives
( ε b − ε a ) ∫ − ∞ ∞ p a ( λ ) v b ( λ ) d λ = B [ p a , v b ] ∣ − ∞ + ∞ . (94) (\varepsilon_b-\varepsilon_a)\int_{-\infty}^{\infty}p_a(\lambda)v_b(\lambda)\,\mathrm{d}\lambda=\left.\mathcal{B}[p_a,v_b]\right|_{-\infty}^{+\infty}. \tag*{(94)} ( ε b − ε a ) ∫ − ∞ ∞ p a ( λ ) v b ( λ ) d λ = B [ p a , v b ] ∣ − ∞ + ∞ . (94) The compactified projective boundary condition identifies the two endpoint values of the concomitant, so the right-hand side vanishes. For distinct eigenvalues this gives the biorthogonality relation
∫ − ∞ ∞ p a ( λ ) v b ( λ ) d λ = g a δ a b , g a : = ∫ − ∞ ∞ p a ( λ ) v a ( λ ) d λ . (95) \int_{-\infty}^{\infty}p_a(\lambda)v_b(\lambda)\,\mathrm{d}\lambda=g_a\delta_{ab},\qquad g_a:=\int_{-\infty}^{\infty}p_a(\lambda)v_a(\lambda)\,\mathrm{d}\lambda. \tag*{(95)} ∫ − ∞ ∞ p a ( λ ) v b ( λ ) d λ = g a δ ab , g a := ∫ − ∞ ∞ p a ( λ ) v a ( λ ) d λ . (95) Thus “biorthogonal” means that the two distinct families { p a } \{p_a\} { p a } { v a } \{v_a\} { v a } p a p_a p a v a v_a v a
It is important that we do not normalize this pairing to unity. As in Eq. (15) of the Main Text, the normalization of each physical right mode has already been fixed by the unit-current condition
B [ p a , v a ] = 1. (96) \mathcal{B}[p_a,v_a]=1. \tag*{(96)} B [ p a , v a ] = 1. (96) Together with the chosen normalization of the polynomial p a p_a p a g a g_a g a (95) are in general not equal to one; they are part of the physical finite-N N N
More generally, without assuming a basis in which the pairing is diagonal, define the Gram matrix
G a b : = ∫ − ∞ ∞ p a ( λ ) v b ( λ ) d λ . (97) G_{ab}:=\int_{-\infty}^{\infty}p_a(\lambda)v_b(\lambda)\,\mathrm{d}\lambda. \tag*{(97)} G ab := ∫ − ∞ ∞ p a ( λ ) v b ( λ ) d λ . (97) Andréief’s identity then gives
∫ R N det a , i [ p a ( λ i ) ] det a , i [ v a ( λ i ) ] d N λ = N ! det G . (98) \int_{\mathbb{R}^N}\det_{a,i}[p_a(\lambda_i)]\det_{a,i}[v_a(\lambda_i)]\,\mathrm{d}^N\lambda=N!\det G. \tag*{(98)} ∫ R N a , i det [ p a ( λ i )] a , i det [ v a ( λ i )] d N λ = N ! det G . (98) Since the product of the two determinants is symmetric under permutations of the λ i \lambda_i λ i C N C_N C N det G \det G det G Z N Z_N Z N (14) of the Main Text is
Z N = det G = ∏ a = 1 N g a (99) Z_N=\det G=\prod_{a=1}^{N}g_a \tag*{(99)} Z N = det G = a = 1 ∏ N g a (99) in the nondegenerate biorthogonal basis. Equivalently, the normalized symmetric extension to R N \mathbb{R}^N R N
P s t s y m ( λ ) = 1 N ! det G det a , i [ p a ( λ i ) ] det a , i [ v a ( λ i ) ] . (100) P_{\mathrm{st}}^{\mathrm{sym}}(\boldsymbol{\lambda})=\frac{1}{N!\det G}\det_{a,i}[p_a(\lambda_i)]\det_{a,i}[v_a(\lambda_i)]. \tag*{(100)} P st sym ( λ ) = N ! det G 1 a , i det [ p a ( λ i )] a , i det [ v a ( λ i )] . (100) This is a biorthogonal ensemble and therefore defines a finite determinantal point process for the λ i \lambda_i λ i [69 , 74 , 75 ] . More precisely, with the standard convention, the correlation functions
R k ( λ 1 , … , λ k ) : = N ! ( N − k ) ! ∫ R N − k P s t s y m ( λ 1 , … , λ N ) ∏ j = k + 1 N d λ j , (101) R_k(\lambda_1,\ldots,\lambda_k):=\frac{N!}{(N-k)!}\int_{\mathbb{R}^{N-k}}P_{\mathrm{st}}^{\mathrm{sym}}(\lambda_1,\ldots,\lambda_N)\prod_{j=k+1}^{N}\mathrm{d}\lambda_j, \tag*{(101)} R k ( λ 1 , … , λ k ) := ( N − k )! N ! ∫ R N − k P st sym ( λ 1 , … , λ N ) j = k + 1 ∏ N d λ j , (101) are all determinants,
R k ( λ 1 , … , λ k ) = det 1 ≤ i , j ≤ k [ K N ( λ i , λ j ) ] . (102) R_k(\lambda_1,\ldots,\lambda_k)=\det_{1\leq i,j\leq k}[K_N(\lambda_i,\lambda_j)]. \tag*{(102)} R k ( λ 1 , … , λ k ) = 1 ≤ i , j ≤ k det [ K N ( λ i , λ j )] . (102) The corresponding finite-rank correlation kernel is
K N ( λ , μ ) = ∑ a , b = 1 N p a ( λ ) ( G − 1 ) b a v b ( μ ) . (103) K_N(\lambda,\mu)=\sum_{a,b=1}^{N}p_a(\lambda)(G^{-1})_{ba}v_b(\mu). \tag*{(103)} K N ( λ , μ ) = a , b = 1 ∑ N p a ( λ ) ( G − 1 ) ba v b ( μ ) . (103) This Gram-matrix form is invariant under invertible changes of the left and right bases and is therefore convenient also at degeneracies. Although the biorthogonal kernel need not be Hermitian, the resulting correlation functions are real and non-negative, as they are the marginals of the physical stationary probability density P s t s y m P_{\mathrm{st}}^{\mathrm{sym}} P st sym G a b = g a δ a b G_{ab}=g_a\delta_{ab} G ab = g a δ ab (103) reduces to
K N ( λ , μ ) = ∑ a = 1 N p a ( λ ) v a ( μ ) g a , (104) K_N(\lambda,\mu)=\sum_{a=1}^{N}\frac{p_a(\lambda)v_a(\mu)}{g_a}, \tag*{(104)} K N ( λ , μ ) = a = 1 ∑ N g a p a ( λ ) v a ( μ ) , (104) In particular, the one-point density of the stationary Dyson–Riccati gas is n ( λ ) = R 1 ( λ ) = K N ( λ , λ ) n(\lambda)=R_1(\lambda)=K_N(\lambda,\lambda) n ( λ ) = R 1 ( λ ) = K N ( λ , λ ) ∫ R n ( λ ) d λ = N \int_{\mathbb{R}}n(\lambda)\,\mathrm{d}\lambda=N ∫ R n ( λ ) d λ = N p a v a ∼ λ − 2 p_av_a\sim\lambda^{-2} p a v a ∼ λ − 2
K N ( λ , λ ) ∼ 1 λ 2 ∑ a = 1 N 1 g a , ∣ λ ∣ → ∞ . (105) K_N(\lambda,\lambda)\sim\frac{1}{\lambda^2}\sum_{a=1}^{N}\frac{1}{g_a},\qquad|\lambda|\to\infty. \tag*{(105)} K N ( λ , λ ) ∼ λ 2 1 a = 1 ∑ N g a 1 , ∣ λ ∣ → ∞. (105) Comparing with the projective-tail identity Eq. (55) gives the exact finite-N N N
ν N ( E ) = ∑ a = 1 N 1 g a , g a : = ∫ − ∞ ∞ p a ( λ ) v a ( λ ) d λ . (106) \nu_N(E)=\sum_{a=1}^{N}\frac{1}{g_a},\qquad g_a:=\int_{-\infty}^{\infty}p_a(\lambda)v_a(\lambda)\,\mathrm{d}\lambda. \tag*{(106)} ν N ( E ) = a = 1 ∑ N g a 1 , g a := ∫ − ∞ ∞ p a ( λ ) v a ( λ ) d λ . (106) The same kernel also gives the total longitudinal volume growth,
Γ N ( E ) = PV ∫ R λ K N ( λ , λ ) d λ . (107) \Gamma_N(E)=\operatorname{PV}\int_{\mathbb{R}}\lambda K_N(\lambda,\lambda)\,\mathrm{d}\lambda. \tag*{(107)} Γ N ( E ) = PV ∫ R λ K N ( λ , λ ) d λ . (107) Thus the finite matrix M N M_N M N M N M_N M N (97) and (S67) provides the regular continuation.
A second, auxiliary determinantal structure will appear below in the cubic-contour representation of Z N ( − ) \mathcal{Z}_N^{(-)} Z N ( − ) K N K_N K N ( p a , v a ) (p_a,v_a) ( p a , v a ) Γ − \Gamma_- Γ −
Fourier–Airy intertwining and the exact Hankel determinant
We now give the detailed derivation of Eq. (21) . The Main Text reduces the N N N
F N ( q ; s ) = det I ( q ) det I ( 0 ) , I a b ( q ) = ∫ R p a ( x ) v b ( x ) e − i q x d x , (108) \mathcal{F}_N(q;s)=\frac{\det I(q)}{\det I(0)},\qquad I_{ab}(q)=\int_{\mathbb{R}}p_a(x)v_b(x)e^{-\mathrm{i}qx}\,\mathrm{d}x, \tag*{(108)} F N ( q ; s ) = det I ( 0 ) det I ( q ) , I ab ( q ) = ∫ R p a ( x ) v b ( x ) e − i q x d x , (108) and the purpose of this subsection is to evaluate the matrix I ( q ) I(q) I ( q ) q > 0 q>0 q > 0 v b v_b v b p b p_b p b e − N V E e^{-NV_E} e − N V E Γ − \Gamma_- Γ − τ N \tau_N τ N
Dimensionless left and right equations. — Introduce
x = λ T 1 / 3 , s = − E T 2 / 3 , e a = ε a T 1 / 3 , (109) x=\frac{\lambda}{T^{1/3}},\qquad s=-\frac{E}{T^{2/3}},\qquad e_a=\frac{\varepsilon_a}{T^{1/3}}, \tag*{(109)} x = T 1/3 λ , s = − T 2/3 E , e a = T 1/3 ε a , (109) and denote the rescaled functions p a ( T 1 / 3 x ) p_a(T^{1/3}x) p a ( T 1/3 x ) v a ( T 1 / 3 x ) v_a(T^{1/3}x) v a ( T 1/3 x ) p a ( x ) p_a(x) p a ( x ) v a ( x ) v_a(x) v a ( x ) ℓ N = T 1 / 3 ℓ ^ N \ell_N=T^{1/3}\widehat{\ell}_N ℓ N = T 1/3 ℓ N (9) becomes
ℓ ^ N = ∂ x 2 − ( x 2 − s ) ∂ x + ( N − 1 ) x , ℓ ^ N p a = e a p a , ℓ ^ N v a = e a v a , ℓ N = ∂ x 2 + ( x 2 − s ) ∂ x + ( N + 1 ) x . (110) \begin{aligned}
\widehat{\ell}_N&=\partial_x^2-(x^2-s)\partial_x+(N-1)x, &
\widehat{\ell}_Np_a&=e_ap_a, &
\widehat{\ell}_Nv_a&=e_av_a,\\
\ell_N&=\partial_x^2+(x^2-s)\partial_x+(N+1)x. \tag*{(110)}
\end{aligned} ℓ N ℓ N = ∂ x 2 − ( x 2 − s ) ∂ x + ( N − 1 ) x , = ∂ x 2 + ( x 2 − s ) ∂ x + ( N + 1 ) x . ℓ N p a = e a p a , ℓ N v a = e a v a , (110) and the weight of the outgoing Kontsevich integral (25) takes the form e − N V E ( λ ) = e − x 3 / 3 + s x e^{-N V_E(\lambda)} = e^{-x^3/3+sx} e − N V E ( λ ) = e − x 3 /3 + s x P N − 1 \mathcal{P}_{N-1} P N − 1
p a ( x ) = ∑ n = 0 N − 1 c a n x n , (111) p_a(x) = \sum_{n=0}^{N-1} c_{an}x^n, \tag*{(111)} p a ( x ) = n = 0 ∑ N − 1 c an x n , (111) on which the left operator acts as
ℓ ^ N x n = n ( n − 1 ) x n − 2 + n s x n − 1 + ( N − 1 − n ) x n + 1 , (112) \hat{\ell}_N x^n = n(n-1)x^{n-2} + nsx^{n-1} + (N-1-n)x^{n+1}, \tag*{(112)} ℓ ^ N x n = n ( n − 1 ) x n − 2 + n s x n − 1 + ( N − 1 − n ) x n + 1 , (112) the rescaled form of Eq. (10) . The last coefficient vanishes at n = N − 1 n=N-1 n = N − 1 P N − 1 \mathcal{P}_{N-1} P N − 1
Fourier equation for a physical right mode.— For a fixed mode b b b
v ~ b ( q ) : = ∫ − ∞ ∞ e − i q x v b ( x ) d x . (113) \widetilde{v}_b(q) := \int_{-\infty}^{\infty} e^{-iqx}v_b(x)\,\mathrm{d}x. \tag*{(113)} v b ( q ) := ∫ − ∞ ∞ e − i q x v b ( x ) d x . (113) The physical boundary condition of Sec. IV B gives v b ( x ) = O ( x − N − 1 ) v_b(x)=O(x^{-N-1}) v b ( x ) = O ( x − N − 1 ) (81) . Hence x m v b ∈ L 1 ( R ) x^m v_b \in L^1(\mathbb{R}) x m v b ∈ L 1 ( R ) 0 ≤ m ≤ N − 1 0 \le m \le N-1 0 ≤ m ≤ N − 1 v ~ b \widetilde{v}_b v b N − 1 N-1 N − 1
∫ R x m v b ( x ) e − i q x d x = ( i ∂ q ) m v ~ b ( q ) , 0 ≤ m ≤ N − 1. (114) \int_{\mathbb{R}} x^m v_b(x)e^{-iqx}\,\mathrm{d}x = (i\partial_q)^m\widetilde{v}_b(q), \qquad0 \le m \le N-1. \tag*{(114)} ∫ R x m v b ( x ) e − i q x d x = ( i ∂ q ) m v b ( q ) , 0 ≤ m ≤ N − 1. (114) The right eigenvalue equation is
ℓ ^ N v b = v b ′ ′ + ( x 2 − s ) v b ′ + ( N + 1 ) x v b = e b v b . (115) \hat{\ell}_N v_b = v_b^{\prime\prime} + (x^2-s)v_b^\prime+ (N+1)xv_b = e_bv_b. \tag*{(115)} ℓ ^ N v b = v b ′′ + ( x 2 − s ) v b ′ + ( N + 1 ) x v b = e b v b . (115) Fourier transformation uses ∂ x ↦ i q \partial_x \mapsto iq ∂ x ↦ i q x ↦ i ∂ q x \mapsto i\partial_q x ↦ i ∂ q (115) , these rules hold for all real q q q q = 0 q=0 q = 0 v b v_b v b v ~ b \widetilde{v}_b v b x m v b x^m v_b x m v b m ≤ N − 1 m \le N-1 m ≤ N − 1 v ~ b \widetilde{v}_b v b N − 1 N-1 N − 1 q = 0 q=0 q = 0 q q q q = 0 q=0 q = 0 q > 0 q>0 q > 0 v ~ b \widetilde{v}_b v b
[ − q 2 − i q ∂ q 2 + i ( N − 1 ) ∂ q − i s q − e b ] v ~ b ( q ) = 0 , q > 0. (116) \left[-q^2-iq\partial_q^2+i(N-1)\partial_q-isq-e_b\right]\widetilde{v}_b(q)=0, \qquad q>0. \tag*{(116)} [ − q 2 − i q ∂ q 2 + i ( N − 1 ) ∂ q − i s q − e b ] v b ( q ) = 0 , q > 0. (116) In the complex variable
z = s − i q , i ∂ q = ∂ z , (117) z=s-iq, \qquad i\partial_q=\partial_z, \tag*{(117)} z = s − i q , i ∂ q = ∂ z , (117) this equation becomes
( D s , N + e b ) v ~ b = 0 , D s , N : = ( z − s ) ( ∂ z 2 − z ) − ( N − 1 ) ∂ z . (118) \left(\mathcal{D}_{s,N}+e_b\right)\widetilde{v}_b=0, \qquad\mathcal{D}_{s,N}:=(z-s)(\partial_z^2-z)-(N-1)\partial_z. \tag*{(118)} ( D s , N + e b ) v b = 0 , D s , N := ( z − s ) ( ∂ z 2 − z ) − ( N − 1 ) ∂ z . (118) For q > 0 q>0 q > 0 z z z z = s z=s z = s
Contour transform of polynomials.— The function e − x 3 / 3 e^{-x^3/3} e − x 3 /3 x x x arg x = 0 \arg x=0 arg x = 0 arg x = ± 2 π / 3 \arg x=\pm2\pi/3 arg x = ± 2 π /3 Γ \Gamma Γ p p p
T Γ [ p ] ( z ) : = ∫ Γ p ( x ) e − x 3 / 3 + z x d x . (119) \mathcal{T}_{\Gamma}[p](z):=\int_{\Gamma}p(x)e^{-x^3/3+zx}\,\mathrm{d}x. \tag*{(119)} T Γ [ p ] ( z ) := ∫ Γ p ( x ) e − x 3 /3 + z x d x . (119) The integral converges for all complex z z z K ( x , z ) = e − x 3 / 3 + z x K(x,z)=e^{-x^3/3+zx} K ( x , z ) = e − x 3 /3 + z x
(i) Multiplication by x x x z z z T Γ [ x p ] = ∂ z T Γ [ p ] \mathcal{T}_{\Gamma}[xp]=\partial_z\mathcal{T}_{\Gamma}[p] T Γ [ x p ] = ∂ z T Γ [ p ] ∂ z K = x K \partial_zK=xK ∂ z K = x K T Γ [ p ] = p ( ∂ z ) F Γ \mathcal{T}_{\Gamma}[p]=p(\partial_z)F_{\Gamma} T Γ [ p ] = p ( ∂ z ) F Γ F Γ : = T Γ [ 1 ] F_{\Gamma}:=\mathcal{T}_{\Gamma}[1] F Γ := T Γ [ 1 ]
(ii) F Γ F_{\Gamma} F Γ ( ∂ z 2 − z ) F Γ = ∫ Γ ( x 2 − z ) K d x = − ∫ Γ ∂ x K d x = 0 (\partial_z^2-z)F_{\Gamma}=\int_{\Gamma}(x^2-z)K\,\mathrm{d}x=-\int_{\Gamma}\partial_xK\,\mathrm{d}x=0 ( ∂ z 2 − z ) F Γ = ∫ Γ ( x 2 − z ) K d x = − ∫ Γ ∂ x K d x = 0 K K K Γ \Gamma Γ F ′ ′ = z F F^{\prime\prime}=zF F ′′ = z F Γ − \Gamma_{-} Γ − e − 2 π i / 3 ∞ e^{-2\pi i/3}\infty e − 2 π i /3 ∞ + ∞ +\infty + ∞ Γ ‾ − \overline{\Gamma}_{-} Γ − e 2 π i / 3 ∞ e^{2\pi i/3}\infty e 2 π i /3 ∞ + ∞ +\infty + ∞ (24) after the substitution w = − x w=-x w = − x C − C_{-} C − Γ − \Gamma_{-} Γ −
F Γ − ( z ) = i π A − ( z ) , F Γ ‾ − ( z ) = − i π A + ( z ) , A ∓ = Ai ∓ i Bi , (120) F_{\Gamma_{-}}(z)=i\pi A_{-}(z),\qquad F_{\overline{\Gamma}_{-}}(z)=-i\pi A_{+}(z),\qquad A_{\mp}=\operatorname{Ai}\mp i\operatorname{Bi}, \tag*{(120)} F Γ − ( z ) = iπ A − ( z ) , F Γ − ( z ) = − iπ A + ( z ) , A ∓ = Ai ∓ i Bi , (120) the second relation following from the first by complex conjugation at real z z z
(iii) Intertwining. Using z K = ∂ x K + x 2 K zK=\partial_xK+x^2K z K = ∂ x K + x 2 K D s , N K = − ∂ x [ ( ∂ x + x 2 − s ) K ] − ( N − 1 ) x K \mathcal{D}_{s,N}K=-\partial_x\left[(\partial_x+x^2-s)K\right]-(N-1)xK D s , N K = − ∂ x [ ( ∂ x + x 2 − s ) K ] − ( N − 1 ) x K p p p
D s , N T Γ [ p ] = − T Γ [ ℓ ~ N † p ] . (121) \mathcal{D}_{s,N}\mathcal{T}_{\Gamma}[p]=-\mathcal{T}_{\Gamma}\left[\widetilde{\ell}^{\dagger}_{N}p\right]. \tag*{(121)} D s , N T Γ [ p ] = − T Γ [ ℓ N † p ] . (121) The operator governing the Fourier transform of the right modes is thus conjugate, through the contour transform, to minus the left operator acting on polynomials. In particular, if p b p_b p b ℓ ~ N † p b = e b p b \widetilde{\ell}^{\dagger}_{N}p_b=e_bp_b ℓ N † p b = e b p b
( D s , N + e b ) T Γ [ p b ] = 0 (122) (\mathcal{D}_{s,N}+e_b)\mathcal{T}_{\Gamma}[p_b]=0 \tag*{(122)} ( D s , N + e b ) T Γ [ p b ] = 0 (122) for every admissible contour Γ \Gamma Γ T Γ [ p b ] ( z ) = ∑ n c b n T Γ [ x n ] ( z ) = ∑ n c b n F Γ ( n ) ( z ) \mathcal{T}_{\Gamma}[p_b](z)=\sum_n c_{bn}\mathcal{T}_{\Gamma}[x^n](z)=\sum_n c_{bn}F_{\Gamma}^{(n)}(z) T Γ [ p b ] ( z ) = ∑ n c bn T Γ [ x n ] ( z ) = ∑ n c bn F Γ ( n ) ( z ) F Γ ( n ) ( z ) = ∂ z n F Γ ( z ) F_{\Gamma}^{(n)}(z)=\partial_z^nF_{\Gamma}(z) F Γ ( n ) ( z ) = ∂ z n F Γ ( z ) F ( n + 2 ) = z F ( n ) + n F ( n − 1 ) F^{(n+2)}=zF^{(n)}+nF^{(n-1)} F ( n + 2 ) = z F ( n ) + n F ( n − 1 ) z F ( n − 1 ) = F ( n + 1 ) − ( n − 1 ) F ( n − 2 ) zF^{(n-1)}=F^{(n+1)}-(n-1)F^{(n-2)} z F ( n − 1 ) = F ( n + 1 ) − ( n − 1 ) F ( n − 2 )
[ ( z − s ) ( ∂ z 2 − z ) − ( N − 1 ) ∂ z ] F ( n ) = − ( N − 1 − n ) F ( n + 1 ) − n s F ( n − 1 ) − n ( n − 1 ) F ( n − 2 ) . (123) \left[(z-s)(\partial_z^2-z)-(N-1)\partial_z\right]F^{(n)}=-(N-1-n)F^{(n+1)}-nsF^{(n-1)}-n(n-1)F^{(n-2)}. \tag*{(123)} [ ( z − s ) ( ∂ z 2 − z ) − ( N − 1 ) ∂ z ] F ( n ) = − ( N − 1 − n ) F ( n + 1 ) − n s F ( n − 1 ) − n ( n − 1 ) F ( n − 2 ) . (123) Comparing with Eq. (S76), we see that the coefficients are exactly the same, with an overall minus sign. The functions T Γ − [ p b ] \mathcal{T}_{\Gamma_{-}}[p_b] T Γ − [ p b ] T Γ ‾ − [ p b ] \mathcal{T}_{\overline{\Gamma}_{-}}[p_b] T Γ − [ p b ] p b ( ∂ z ) ( α F Γ − + β F Γ ‾ − ) = 0 p_b(\partial_z)(\alpha F_{\Gamma_{-}}+\beta F_{\overline{\Gamma}_{-}})=0 p b ( ∂ z ) ( α F Γ − + β F Γ − ) = 0 N − 1 N-1 N − 1 q > 0 q>0 q > 0
Selection of the outgoing contour. Since v b ∈ L 1 ( R ) v_b\in L^1(\mathbb{R}) v b ∈ L 1 ( R )
v ~ b ( q ) ⟶ 0 , q → + ∞ . (124) \widetilde{v}_b(q)\longrightarrow0,\qquad q\to+\infty. \tag*{(124)} v b ( q ) ⟶ 0 , q → + ∞. (124) In the contour representation the behaviour at large q q q z = s − i q z=s-iq z = s − i q e − i q x e^{-iqx} e − i q x e q Im x e^{q\operatorname{Im}x} e q Im x Γ − \Gamma_{-} Γ − Γ − − i c \Gamma_{-}-ic Γ − − i c c > 0 c>0 c > 0 Im x ≤ − c \operatorname{Im}x\leq-c Im x ≤ − c ∣ T Γ − [ p ] ( s − i q ) ∣ ≤ C c e − c q |\mathcal{T}_{\Gamma_{-}}[p](s-iq)|\leq C_ce^{-cq} ∣ T Γ − [ p ] ( s − i q ) ∣ ≤ C c e − c q c > 0 c>0 c > 0 Γ − \Gamma_{-} Γ − A − ( z ) = 2 e − i π / 3 Ai ( z e 2 π i / 3 ) A_{-}(z)=2e^{-i\pi/3}\operatorname{Ai}(ze^{2\pi i/3}) A − ( z ) = 2 e − iπ /3 Ai ( z e 2 π i /3 ) arg ( z e 2 π i / 3 ) = π / 6 \arg(ze^{2\pi i/3})=\pi/6 arg ( z e 2 π i /3 ) = π /6 [52 ] . By contrast, Γ ‾ − \overline{\Gamma}_{-} Γ − e 2 π i / 3 ∞ e^{2\pi i/3}\infty e 2 π i /3 ∞ A + ( z ) = 2 e i π / 3 Ai ( z e − 2 π i / 3 ) A_{+}(z)=2e^{i\pi/3}\operatorname{Ai}(ze^{-2\pi i/3}) A + ( z ) = 2 e iπ /3 Ai ( z e − 2 π i /3 ) arg = 5 π / 6 \arg=5\pi/6 arg = 5 π /6 p b ( ∂ z ) A + p_b(\partial_z)A_{+} p b ( ∂ z ) A + p b p_b p b
v ~ b ( q ) = η b T Γ − [ p b ] ( s − i q ) = η b ∫ Γ − p b ( x ) e − x 3 / 3 + s x e − i q x d x , q > 0. (125) \widetilde{v}_b(q)=\eta_b\mathcal{T}_{\Gamma_{-}}[p_b](s-iq)=\eta_b\int_{\Gamma_{-}}p_b(x)e^{-x^3/3+sx}e^{-iqx}\,dx,\qquad q>0. \tag*{(125)} v b ( q ) = η b T Γ − [ p b ] ( s − i q ) = η b ∫ Γ − p b ( x ) e − x 3 /3 + s x e − i q x d x , q > 0. (125) where η b \eta_b η b q q q η b ≠ 0 \eta_b\ne0 η b = 0 v b ≠ 0 v_b\ne0 v b = 0 v b v_b v b
Overlap matrix and the Hankel determinant. Combining Eqs. (S78), (S81) and (S89) with property (i) of the contour transform,
I a b ( q ) = ∑ m = 0 N − 1 c a m ∂ z m v ~ b ( q ) = η b T Γ − [ p a p b ] ( z ) = η b ∫ Γ − p a ( x ) p b ( x ) e − x 3 / 3 + z x d x , z = s − i q . (126) I_{ab}(q)=\sum_{m=0}^{N-1}c_{am}\partial_z^m\widetilde{v}_b(q)=\eta_b\mathcal{T}_{\Gamma_{-}}[p_ap_b](z)=\eta_b\int_{\Gamma_{-}}p_a(x)p_b(x)e^{-x^3/3+zx}\,dx,\qquad z=s-iq. \tag*{(126)} I ab ( q ) = m = 0 ∑ N − 1 c am ∂ z m v b ( q ) = η b T Γ − [ p a p b ] ( z ) = η b ∫ Γ − p a ( x ) p b ( x ) e − x 3 /3 + z x d x , z = s − i q . (126) The overlap matrix is therefore the symmetric bilinear product (S121) of the orthogonal-polynomial problem, evaluated at the complex-shifted value z = s − i q z=s-iq z = s − i q s s s E + i q T 2 / 3 E+iqT^{2/3} E + i q T 2/3
I ( q ) = C M ( z ) C T E , M m n ( z ) = μ m + n ( z ) , μ k ( z ) : = ∫ Γ − x k e − x 3 / 3 + z x d x = i π A − ( k ) ( z ) , (127) I(q)=C\,M(z)\,C^{\mathsf T}E,\qquad M_{mn}(z)=\mu_{m+n}(z),\qquad\mu_k(z):=\int_{\Gamma_-}x^k e^{-x^3/3+zx}\,\mathrm{d}x=i\pi\,\mathcal{A}^{(k)}_-(z), \tag*{(127)} I ( q ) = C M ( z ) C T E , M mn ( z ) = μ m + n ( z ) , μ k ( z ) := ∫ Γ − x k e − x 3 /3 + z x d x = iπ A − ( k ) ( z ) , (127) with C a n = c a n C_{an}=c_{an} C an = c an E = diag ( η 1 , … , η N ) E=\operatorname{diag}(\eta_1,\ldots,\eta_N) E = diag ( η 1 , … , η N ) m + n m+n m + n N N N P N − 1 \mathcal{P}_{N-1} P N − 1 C C C
det I ( q ) = ( det C ) 2 det E det 0 ≤ m , n ≤ N − 1 [ μ m + n ( z ) ] = ( i π ) N ( det C ) 2 det E τ N ( s − i q ) , (128) \det I(q)=(\det C)^2\det E\det_{0\le m,n\le N-1}[\mu_{m+n}(z)]=(i\pi)^N(\det C)^2\det E\,\tau_N(s-iq), \tag*{(128)} det I ( q ) = ( det C ) 2 det E 0 ≤ m , n ≤ N − 1 det [ μ m + n ( z )] = ( iπ ) N ( det C ) 2 det E τ N ( s − i q ) , (128) with τ N \tau_N τ N (22) . Equivalently, by Andréief’s identity, the Hankel determinant of the moments is the N N N 1 N ! ∫ Γ − N V N ( x ) 2 ∏ j e − x j 3 / 3 + z x j d N x \frac{1}{N!}\int_{\Gamma_-^N}V_N(x)^2\prod_j e^{-x_j^3/3+zx_j}\,\mathrm{d}^N x N ! 1 ∫ Γ − N V N ( x ) 2 ∏ j e − x j 3 /3 + z x j d N x (25) at the shifted energy. The matrices C C C E E E q q q q → 0 + q\to0+ q → 0 +
det I ( 0 ) = ( i π ) N ( det C ) 2 det E τ N ( s ) , (129) \det I(0)=(i\pi)^N(\det C)^2\det E\,\tau_N(s), \tag*{(129)} det I ( 0 ) = ( iπ ) N ( det C ) 2 det E τ N ( s ) , (129) and all basis-dependent factors cancel in the ratio,
F N ( q ; s ) = τ N ( s − i q ) τ N ( s ) = Z N ( − ) ( E + i q T 2 / 3 ) Z N ( − ) ( E ) , q > 0. (130) \mathcal{F}_N(q;s)=\frac{\tau_N(s-iq)}{\tau_N(s)}=\frac{\mathcal{Z}^{(-)}_N(E+iqT^{2/3})}{\mathcal{Z}^{(-)}_N(E)},\qquad q>0. \tag*{(130)} F N ( q ; s ) = τ N ( s ) τ N ( s − i q ) = Z N ( − ) ( E ) Z N ( − ) ( E + i q T 2/3 ) , q > 0. (130) This proves Eq. (21) .
Equation (S93) has a useful corollary. Up to the factor T N / 3 T^{N/3} T N /3 det I ( 0 ) \det I(0) det I ( 0 ) (100) . Since this density is a probability density, det G ≠ 0 \det G\ne0 det G = 0
τ N ( s ) ≠ 0 for all real s . (131) \tau_N(s)\ne0\qquad\text{for all real }s. \tag*{(131)} τ N ( s ) = 0 for all real s . (131) Equivalently, the outgoing Kontsevich integral Z N ( − ) ∝ τ N ( − E / T 2 / 3 ) \mathcal{Z}^{(-)}_N\propto\tau_N(-E/T^{2/3}) Z N ( − ) ∝ τ N ( − E / T 2/3 ) log Z N ( − ) \log\mathcal{Z}^{(-)}_N log Z N ( − ) Ξ N = Γ N − i π ν N \Xi_N=\Gamma_N-i\pi\nu_N Ξ N = Γ N − iπ ν N E E E
The derivation was written for generic parameters, where the polynomial eigenvalues are nondegenerate and the eigenvectors form an ordinary basis. The determinant ratio itself is basis independent, however, and the physical stationary process varies continuously with s s s (131) , therefore extend through isolated degeneracies by continuity; equivalently, one may work there with a Jordan (confluent) polynomial basis, in which the same determinant identity is obtained directly.
Check at N = 1 N=1 N = 1 For one channel the polynomial sector contains only p 1 = 1 p_1=1 p 1 = 1 e 1 = 0 e_1=0 e 1 = 0 (118) reduces to the Airy equation itself. Equation (S89) gives v ~ e 1 ( q ) = i π η 1 A − ( s − i q ) \widetilde v_{e_1}(q)=i\pi\eta_1\mathcal{A}_-(s-iq) v e 1 ( q ) = iπ η 1 A − ( s − i q )
F 1 ( q ; s ) = A − ( s − i q ) A − ( s ) , (132) \mathcal{F}_1(q;s)=\frac{\mathcal{A}_-(s-iq)}{\mathcal{A}_-(s)}, \tag*{(132)} F 1 ( q ; s ) = A − ( s ) A − ( s − i q ) , (132) which is Halperin’s result [2 ] , since τ 1 = A − \tau_1=\mathcal{A}_- τ 1 = A − N N N
Explicit illustration for N = 2 N=2 N = 2 For N = 2 N=2 N = 2 P 1 = span { 1 , x } \mathcal{P}_1=\operatorname{span}\{1,x\} P 1 = span { 1 , x } (112) gives e ~ 1 2 = 1 \widetilde e_1^2=1 e 1 2 = 1 e ~ 2 2 = s \widetilde e_2^2=s e 2 2 = s p 1 = c 0 + c 1 x p_1=c_0+c_1x p 1 = c 0 + c 1 x ℓ ~ 2 p = e p \widetilde{\ell}_2p=ep ℓ 2 p = e p s c 1 = e c 0 sc_1=ec_0 s c 1 = e c 0 c 0 = e c 1 c_0=ec_1 c 0 = e c 1 e 2 = s e^2=s e 2 = s s = 0 s=0 s = 0 s \sqrt{s} s
e ± = ± s , p ± ( x ) = x ± s , C = ( s 1 − s 1 ) , ( det C ) 2 = 4 s . (133) e_\pm=\pm\sqrt{s},\qquad p_\pm(x)=x\pm\sqrt{s},\qquad C=\begin{pmatrix}\sqrt{s}&1\\-\sqrt{s}&1\end{pmatrix},\qquad(\det C)^2=4s. \tag*{(133)} e ± = ± s , p ± ( x ) = x ± s , C = ( s − s 1 1 ) , ( det C ) 2 = 4 s . (133) The final result depends only on s s s (125) the Fourier transforms of the two physical right modes are v ~ ± = η ± ( μ 1 ± s μ 0 ) \widetilde v_\pm=\eta_\pm(\mu_1\pm\sqrt{s}\,\mu_0) v ± = η ± ( μ 1 ± s μ 0 ) (126) gives I a b = η b [ μ 2 + ( e a + e b ) μ 1 + e a e b μ 0 ] I_{ab}=\eta_b[\mu_2+(e_a+e_b)\mu_1+e_ae_b\mu_0] I ab = η b [ μ 2 + ( e a + e b ) μ 1 + e a e b μ 0 ] μ 2 = z μ 0 \mu_2=z\mu_0 μ 2 = z μ 0
I ( q ) = ( ( z + s ) μ 0 + 2 s μ 1 ( z − s ) μ 0 ( z − s ) μ 0 ( z + s ) μ 0 − 2 s μ 1 ) ( η + 0 0 η − ) , (134) I(q)=\begin{pmatrix}(z+s)\mu_0+2\sqrt{s}\,\mu_1&(z-s)\mu_0\\(z-s)\mu_0&(z+s)\mu_0-2\sqrt{s}\,\mu_1\end{pmatrix}\begin{pmatrix}\eta_+&0\\0&\eta_-\end{pmatrix}, \tag*{(134)} I ( q ) = ( ( z + s ) μ 0 + 2 s μ 1 ( z − s ) μ 0 ( z − s ) μ 0 ( z + s ) μ 0 − 2 s μ 1 ) ( η + 0 0 η − ) , (134) with all moments evaluated at z = s − i q z=s-iq z = s − i q
det I ( q ) = 4 s η + η − ( z μ 0 2 − μ 1 2 ) = 4 s η + η − det ( μ 0 μ 1 μ 1 μ 2 ) = ( i π ) 2 4 s η + η − τ 2 ( s − i q ) , (135) \det I(q)=4s\eta_{+}\eta_{-}\left(z\mu_{0}^{2}-\mu_{1}^{2}\right)=4s\eta_{+}\eta_{-}\det\begin{pmatrix}\mu_{0}&\mu_{1}\\\mu_{1}&\mu_{2}\end{pmatrix}=(i\pi)^2 4s\eta_{+}\eta_{-}\tau_{2}(s-iq), \tag*{(135)} det I ( q ) = 4 s η + η − ( z μ 0 2 − μ 1 2 ) = 4 s η + η − det ( μ 0 μ 1 μ 1 μ 2 ) = ( iπ ) 2 4 s η + η − τ 2 ( s − i q ) , (135) where τ 2 ( z ) = z A − 2 ( z ) − [ A − ′ ( z ) ] 2 \tau_{2}(z)=zA_{-}^{2}(z)-[A_{-}'(z)]^{2} τ 2 ( z ) = z A − 2 ( z ) − [ A − ′ ( z ) ] 2 N = 2 N=2 N = 2 (128) , with ( det C ) 2 = 4 s (\det C)^2=4s ( det C ) 2 = 4 s q q q q = 0 q=0 q = 0
F 2 ( q ; s ) = ( s − i q ) A − 2 ( s − i q ) − [ A − ′ ( s − i q ) ] 2 s A − 2 ( s ) − [ A − ′ ( s ) ] 2 , q > 0. (136) F_{2}(q;s)=\frac{(s-iq)A_{-}^{2}(s-iq)-[A_{-}'(s-iq)]^{2}}{sA_{-}^{2}(s)-[A_{-}'(s)]^{2}},\qquad q>0. \tag*{(136)} F 2 ( q ; s ) = s A − 2 ( s ) − [ A − ′ ( s ) ] 2 ( s − i q ) A − 2 ( s − i q ) − [ A − ′ ( s − i q ) ] 2 , q > 0. (136) At s = 0 s=0 s = 0 p + p_{+} p + ( det C ) 2 = 4 s → 0 (\det C)^2=4s\to0 ( det C ) 2 = 4 s → 0
Quartic Schrödinger representation and the large-N N N
Schrödinger gauge and projective boundary conditions
The same one-body problem admits a useful Schrödinger representation. Set
g ( λ ) = U E ( λ ) 2 T , ϕ a ( λ ) = e − g ( λ ) p a ( λ ) , χ a ( λ ) = e g ( λ ) v a ( λ ) . (137) g(\lambda)=\frac{U_E(\lambda)}{2T},\qquad\phi_a(\lambda)=e^{-g(\lambda)}p_a(\lambda),\qquad\chi_a(\lambda)=e^{g(\lambda)}v_a(\lambda). \tag*{(137)} g ( λ ) = 2 T U E ( λ ) , ϕ a ( λ ) = e − g ( λ ) p a ( λ ) , χ a ( λ ) = e g ( λ ) v a ( λ ) . (137) The left and right equations then become the same second-order differential equation,
H N ϕ a = − ϵ a ϕ a , H N χ a = − ϵ a χ a , (138) H_N\phi_a=-\epsilon_a\phi_a,\qquad H_N\chi_a=-\epsilon_a\chi_a, \tag*{(138)} H N ϕ a = − ϵ a ϕ a , H N χ a = − ϵ a χ a , (138) with quartic differential expression
H N = − T ∂ λ 2 + ( E + λ 2 ) 2 4 T − N λ . (139) H_N=-T\partial_\lambda^2+\frac{(E+\lambda^2)^2}{4T}-N\lambda. \tag*{(139)} H N = − T ∂ λ 2 + 4 T ( E + λ 2 ) 2 − N λ . (139) Moreover the unit concomitant becomes the constant Wronskian
T ( ϕ a χ a ′ − ϕ a ′ χ a ) = 1. (140) T(\phi_a\chi_a'-\phi_a'\chi_a)=1. \tag*{(140)} T ( ϕ a χ a ′ − ϕ a ′ χ a ) = 1. (140) It is important that Eq. (139) specifies a differential expression, not by itself an L 2 ( R ) L^2(\mathbb{R}) L 2 ( R )
ϕ a ( λ ) ∼ λ N − 1 e − U E ( λ ) / ( 2 T ) , χ a ( λ ) ∼ λ − N − 1 e U E ( λ ) / ( 2 T ) , ∣ λ ∣ → ∞ , (141) \phi_a(\lambda)\sim\lambda^{N-1}e^{-U_E(\lambda)/(2T)},\qquad\chi_a(\lambda)\sim\lambda^{-N-1}e^{U_E(\lambda)/(2T)},\qquad|\lambda|\to\infty, \tag*{(141)} ϕ a ( λ ) ∼ λ N − 1 e − U E ( λ ) / ( 2 T ) , χ a ( λ ) ∼ λ − N − 1 e U E ( λ ) / ( 2 T ) , ∣ λ ∣ → ∞ , (141) with the two functions decaying at opposite ends of the real axis. Thus the algebraic levels − ϵ a -\epsilon_a − ϵ a N N N
Semiclassical filled Fermi sea
Now take N → ∞ N\to\infty N → ∞ J J J T = J 2 / ( 2 N ) T=J^2/(2N) T = J 2 / ( 2 N ) (139) can be written
H N = N [ − J 2 2 N 2 ∂ λ 2 + W E ( λ ) ] , W E ( λ ) = ( E + λ 2 ) 2 2 J 2 − λ . (142) H_N=N\left[-\frac{J^2}{2N^2}\partial_\lambda^2+W_E(\lambda)\right],\qquad W_E(\lambda)=\frac{(E+\lambda^2)^2}{2J^2}-\lambda. \tag*{(142)} H N = N [ − 2 N 2 J 2 ∂ λ 2 + W E ( λ ) ] , W E ( λ ) = 2 J 2 ( E + λ 2 ) 2 − λ . (142) The effective semiclassical parameter is therefore ℏ e f f = J / N \hbar_{\mathrm{eff}}=J/N ℏ eff = J / N N N N p 2 / 2 + W E ( λ ) p^2/2+W_E(\lambda) p 2 /2 + W E ( λ ) μ F \mu_F μ F
N = 1 2 π ℏ e f f ∬ p 2 / 2 + W E ( λ ) ≤ μ F d p d λ = N 2 π J ∬ p 2 / 2 + W E ( λ ) ≤ μ F d p d λ ⟹ 1 = 1 2 π J ∬ p 2 / 2 + W E ( λ ) ≤ μ F d p d λ . (143) N=\frac{1}{2\pi\hbar_{\mathrm{eff}}}\iint_{p^2/2+W_E(\lambda)\leq\mu_F}dp\,d\lambda=\frac{N}{2\pi J}\iint_{p^2/2+W_E(\lambda)\leq\mu_F}dp\,d\lambda\quad\Longrightarrow\quad1=\frac{1}{2\pi J}\iint_{p^2/2+W_E(\lambda)\leq\mu_F}dp\,d\lambda. \tag*{(143)} N = 2 π ℏ eff 1 ∬ p 2 /2 + W E ( λ ) ≤ μ F d p d λ = 2 π J N ∬ p 2 /2 + W E ( λ ) ≤ μ F d p d λ ⟹ 1 = 2 π J 1 ∬ p 2 /2 + W E ( λ ) ≤ μ F d p d λ . (143) Figure S1. Evolution of the tilted quartic potential w ϵ ( x ) = W E ( λ ) / s J = ( x 2 + ϵ ) 2 − x w_{\epsilon}(x) = W_E(\lambda)/s_J = (x^2+\epsilon)^2-x w ϵ ( x ) = W E ( λ ) / s J = ( x 2 + ϵ ) 2 − x x = λ / s J x=\lambda/s_J x = λ / s J ϵ = E / s J 2 \epsilon=E/s_J^2 ϵ = E / s J 2 x ∗ = − 1 / 2 x_*=-1/2 x ∗ = − 1/2 w ′ = w ′ ′ = 0 w'=w''=0 w ′ = w ′′ = 0 w ′ ′ ′ ≠ 0 w'''\ne0 w ′′′ = 0 3 / 4 3/4 3/4 [ − 1 / 2 , 3 / 2 ] [-1/2,3/2] [ − 1/2 , 3/2 ]
The exact trace identity (S38) says that the sum of the filled algebraic one-body energies − ϵ a -\epsilon_a − ϵ a
1 2 π J ∬ p 2 / 2 + W E ( λ ) ≤ μ F [ p 2 2 + W E ( λ ) ] d p d λ = 0. \frac{1}{2\pi J}\iint_{p^2/2+W_E(\lambda)\leq\mu_F}\left[\frac{p^2}{2}+W_E(\lambda)\right]\,\mathrm{d}p\,\mathrm{d}\lambda=0. 2 π J 1 ∬ p 2 /2 + W E ( λ ) ≤ μ F [ 2 p 2 + W E ( λ ) ] d p d λ = 0. In the confined regime E < E ∗ E<E^* E < E ∗ W E W_E W E λ + ( E ) > 0 \lambda_+(E)>0 λ + ( E ) > 0 λ − ( E ) < 0 \lambda_-(E)<0 λ − ( E ) < 0
Let λ − ( E ) \lambda_-(E) λ − ( E )
W E ′ ( λ − ) = 0 , 2 λ − ( E + λ − 2 ) = J 2 . W'_E(\lambda_-)=0,\qquad2\lambda_-(E+\lambda_-^2)=J^2. W E ′ ( λ − ) = 0 , 2 λ − ( E + λ − 2 ) = J 2 . The Fermi level coincides exactly with the value of the potential at this higher minimum,
μ F = W E ( λ − ) = J 2 8 λ − 2 − λ − . (144) \mu_F=W_E(\lambda_-)=\frac{J^2}{8\lambda_-^2}-\lambda_-. \tag*{(144)} μ F = W E ( λ − ) = 8 λ − 2 J 2 − λ − . (144) Indeed, setting a 2 : = − J 2 / λ − > 0 a^2:=-J^2/\lambda_->0 a 2 := − J 2 / λ − > 0
2 J 2 [ W E ( λ ) − μ F ] = ( λ − λ − ) 2 [ ( λ + λ − ) 2 − a 2 ] . (145) 2J^2[W_E(\lambda)-\mu_F]=(\lambda-\lambda_-)^2\left[(\lambda+\lambda_-)^2-a^2\right]. \tag*{(145)} 2 J 2 [ W E ( λ ) − μ F ] = ( λ − λ − ) 2 [ ( λ + λ − ) 2 − a 2 ] . (145) The occupied component of the Fermi sea is therefore the compact interval
− λ − − a ≤ λ ≤ − λ − + a , -\lambda_- -a\leq\lambda\leq-\lambda_-+a, − λ − − a ≤ λ ≤ − λ − + a , lying in the deeper right-hand well of W E W_E W E p p p (145) , gives
∬ o c c d p d λ = 2 ∫ − λ − − a − λ − + a 2 [ μ F − W E ( λ ) ] d λ = 2 π J . (146) \iint_{\mathrm{occ}}\mathrm{d}p\,\mathrm{d}\lambda
=2\int_{-\lambda_- -a}^{-\lambda_-+a}\sqrt{2[\mu_F-W_E(\lambda)]}\,\mathrm{d}\lambda
=2\pi J. \tag*{(146)} ∬ occ d p d λ = 2 ∫ − λ − − a − λ − + a 2 [ μ F − W E ( λ )] d λ = 2 π J . (146) so Eq. (143) is satisfied exactly; the same factorization also gives the zero mean energy above. Thus the state-counting and trace conditions select the same Fermi level.
Thus, at leading semiclassical order the particles occupy the deeper right-hand well, although the exact finite-N N N L 2 L^2 L 2 λ − → λ ∗ = − s J / 2 \lambda_-\to\lambda_*=-s_J/2 λ − → λ ∗ = − s J /2 μ F → μ F ∗ = 3 s J / 4 \mu_F\to\mu_F^*=3s_J/4 μ F → μ F ∗ = 3 s J /4
Stieltjes roots and the Allez–Dumaz planar solution
The same planar limit is visible directly in the polynomial roots. For a sequence of modes, introduce the normalized empirical root measure
μ a ( N ) : = 1 N − 1 ∑ j = 1 N − 1 δ r j ( a ) . (147) \mu_a^{(N)} := \frac{1}{N-1}\sum_{j=1}^{N-1}\delta_{r_j^{(a)}}. \tag*{(147)} μ a ( N ) := N − 1 1 j = 1 ∑ N − 1 δ r j ( a ) . (147) The roots need not be real at finite N N N N → ∞ N \to\infty N → ∞ Σ \Sigma Σ ϱ ( r ) d r \varrho(r)\,dr ϱ ( r ) d r 2 T = J 2 / N 2T=J^2/N 2 T = J 2 / N (77) gives
λ 2 + E = J 2 PV ∫ Σ ϱ ( r ) λ − r d r , ∫ Σ ϱ ( r ) d r = 1. (148) \lambda^2+E=J^2\operatorname{PV}\int_{\Sigma}\frac{\varrho(r)}{\lambda-r}\,dr,\qquad\int_{\Sigma}\varrho(r)\,dr=1. \tag*{(148)} λ 2 + E = J 2 PV ∫ Σ λ − r ϱ ( r ) d r , ∫ Σ ϱ ( r ) d r = 1. (148) This is precisely the planar Coulomb-gas saddle equation underlying the nonconfining cubic Dyson model of Allez and Dumaz [34 ] . The exact roots of the finite-N N N
It is useful to recall the corresponding planar solution more explicitly. To avoid confusion with the spectral DOS ρ N ( E ) \rho_N(E) ρ N ( E ) σ E ( λ ) \sigma_E(\lambda) σ E ( λ ) N N N N N N n ( λ ) n(\lambda) n ( λ ) N N N σ E ( λ ) = lim N → ∞ N − 1 n ( λ ) \sigma_E(\lambda)=\lim_{N\to\infty}N^{-1}n(\lambda) σ E ( λ ) = lim N → ∞ N − 1 n ( λ ) G ( z ) G(z) G ( z ) G ( z ) ∼ z − 1 G(z)\sim z^{-1} G ( z ) ∼ z − 1 z → ∞ z\to\infty z → ∞
G ( z ) = 1 J 2 [ z 2 + E − ( z + d ) ( z − Λ − ) ( z − Λ + ) ] , (149) G(z)=\frac{1}{J^2}\left[z^2+E-(z+d)\sqrt{(z-\Lambda_-)(z-\Lambda_+)}\right], \tag*{(149)} G ( z ) = J 2 1 [ z 2 + E − ( z + d ) ( z − Λ − ) ( z − Λ + ) ] , (149) Λ ± = d ± J d , d 3 + E d + J 2 2 = 0 , \Lambda_\pm=d\pm\frac{J}{\sqrt{d}},\qquad d^3+Ed+\frac{J^2}{2}=0, Λ ± = d ± d J , d 3 + E d + 2 J 2 = 0 , where the square-root branch is fixed by G ( z ) ∼ z − 1 G(z)\sim z^{-1} G ( z ) ∼ z − 1 d ( E ) d(E) d ( E ) d ( E ) ∼ − i E d(E)\sim-\mathrm{i}\sqrt{E} d ( E ) ∼ − i E E E E
In the confined phase E < E ∗ E<E^* E < E ∗ (149) ; the cut is the real interval [ Λ − , Λ + ] [\Lambda_-,\Lambda_+] [ Λ − , Λ + ]
σ E c o n f ( λ ) = λ + d π J 2 ( Λ + − λ ) ( λ − Λ − ) 1 [ Λ − , Λ + ] ( λ ) . (150) \sigma_E^{\mathrm{conf}}(\lambda)=\frac{\lambda+d}{\pi J^2}\sqrt{(\Lambda_+-\lambda)(\lambda-\Lambda_-)}\,\mathbf{1}_{[\Lambda_-,\Lambda_+]}(\lambda). \tag*{(150)} σ E conf ( λ ) = π J 2 λ + d ( Λ + − λ ) ( λ − Λ − ) 1 [ Λ − , Λ + ] ( λ ) . (150) Thus in this phase the planar solution is an ordinary equilibrium Coulomb gas on a compact real interval. Notice that this is the same confined regime in which the Thomas–Fermi sea of § 0.9 B is trapped in the deeper well of W E W_E W E
For E > E ∗ E>E^* E > E ∗ d ( E ) d(E) d ( E )
σ E f l o w ( λ ) = − 1 π Im G ( λ + i 0 ) = 1 π J 2 Im [ ( λ + d ) ( λ − d ) 2 − J 2 d ] , λ ∈ R , (151) \sigma_E^{\mathrm{flow}}(\lambda)=-\frac{1}{\pi}\operatorname{Im}G(\lambda+\mathrm{i}0)=\frac{1}{\pi J^2}\operatorname{Im}\left[(\lambda+d)\sqrt{(\lambda-d)^2-\frac{J^2}{d}}\right],\qquad\lambda\in\mathbb{R}, \tag*{(151)} σ E flow ( λ ) = − π 1 Im G ( λ + i 0 ) = π J 2 1 Im [ ( λ + d ) ( λ − d ) 2 − d J 2 ] , λ ∈ R , (151) The algebraic tails follow by expanding the square root on the inherited branch:
( λ + d ) ( λ − d ) 2 − J 2 d = λ 2 + E − J 2 λ − J 2 λ 2 ( d + J 2 8 d 2 ) + O ( λ − 3 ) . (\lambda+d)\sqrt{(\lambda-d)^2-\frac{J^2}{d}}=\lambda^2+E-\frac{J^2}{\lambda}-\frac{J^2}{\lambda^2}\left(d+\frac{J^2}{8d^2}\right)+O(\lambda^{-3}). ( λ + d ) ( λ − d ) 2 − d J 2 = λ 2 + E − λ J 2 − λ 2 J 2 ( d + 8 d 2 J 2 ) + O ( λ − 3 ) . The first three terms are real, hence
σ E f l o w ( λ ) ∼ j E λ 2 , j E = − 1 π Im ( d + J 2 8 d 2 ) > 0 , λ → ± ∞ . \sigma_E^{\mathrm{flow}}(\lambda)\sim\frac{j_E}{\lambda^2},\qquad j_E=-\frac{1}{\pi}\operatorname{Im}\left(d+\frac{J^2}{8d^2}\right)>0,\qquad\lambda\to\pm\infty. σ E flow ( λ ) ∼ λ 2 j E , j E = − π 1 Im ( d + 8 d 2 J 2 ) > 0 , λ → ± ∞. Here j E = lim N → ∞ ν N ( E ) / N j_E=\lim_{N\to\infty}\nu_N(E)/N j E = lim N → ∞ ν N ( E ) / N
For the matrix Schrödinger problem, the confined and flowing branches correspond respectively to the region below the limiting spectral edge and to the spectral bulk above it [28 , 30 ] .
The Thomas–Fermi and electrostatic pictures are therefore two representations of the same planar algebraic sector: Eq. (148) uses the zeros of the exact polynomial modes, whereas Eq. (143) uses the semiclassical phase space of their quartic-gauge representatives. In the confined phase both reduce to a compact real saddle; beyond the transition the physical Riccati problem selects its current-carrying analytic continuation.
Critical point, critical density, and semiclassical scaling
The collective edge is particularly transparent in the quartic Schrödinger representation. From Eq. (142) ,
W E ′ ( λ ) = 2 λ ( E + λ 2 ) J 2 − 1 , W E ′ ′ ( λ ) = 2 ( E + 3 λ 2 ) J 2 , W E ′ ( λ ) = 0 ⟺ 2 λ 3 + 2 E λ − J 2 = 0. (152) W'_E(\lambda)=\frac{2\lambda(E+\lambda^2)}{J^2}-1,\qquad W''_E(\lambda)=\frac{2(E+3\lambda^2)}{J^2},\qquad W'_E(\lambda)=0\Longleftrightarrow2\lambda^3+2E\lambda-J^2=0. \tag*{(152)} W E ′ ( λ ) = J 2 2 λ ( E + λ 2 ) − 1 , W E ′′ ( λ ) = J 2 2 ( E + 3 λ 2 ) , W E ′ ( λ ) = 0 ⟺ 2 λ 3 + 2 E λ − J 2 = 0. (152) The transition occurs when the secondary minimum introduced in § 0.9 B merges with the intervening maximum, so W E ′ ( λ ∗ ) = W E ′ ′ ( λ ∗ ) = 0 W'_E(\lambda_*)=W''_E(\lambda_*)=0 W E ′ ( λ ∗ ) = W E ′′ ( λ ∗ ) = 0 s J = ( 2 J 2 ) 1 / 3 s_J=(2J^2)^{1/3} s J = ( 2 J 2 ) 1/3
λ ∗ = − s J 2 , E ∗ = − 3 s J 2 4 , μ F ∗ = W E ∗ ( λ ∗ ) = 3 s J 4 . (153) \lambda_*=-\frac{s_J}{2},\qquad E_*=-\frac{3s_J^2}{4},\qquad\mu_F^*=W_{E_*}(\lambda_*)=\frac{3s_J}{4}. \tag*{(153)} λ ∗ = − 2 s J , E ∗ = − 4 3 s J 2 , μ F ∗ = W E ∗ ( λ ∗ ) = 4 3 s J . (153) The last value is the critical limit of the Fermi level found in Eq. (144) .
Comparing the extremum equation in Eq. (152) with the cubic equation for d d d (149) identifies d = − λ d=-\lambda d = − λ d = − λ − ( E ) d=-\lambda_-(E) d = − λ − ( E ) λ − ( E ) \lambda_-(E) λ − ( E ) W E W_E W E § 0.9 B. At criticality d ∗ = s J / 2 d_*=s_J/2 d ∗ = s J /2 (149) become Λ − ∗ = − s J / 2 = λ ∗ \Lambda_-^*=-s_J/2=\lambda_* Λ − ∗ = − s J /2 = λ ∗ Λ + ∗ = 3 s J / 2 \Lambda_+^*=3s_J/2 Λ + ∗ = 3 s J /2 (150) therefore gives
σ ∗ ( λ ) = 1 π J 2 ( λ + s J 2 ) 3 / 2 ( 3 s J 2 − λ ) 1 / 2 1 [ − s J / 2 , 3 s J / 2 ] ( λ ) . (154) \sigma_*(\lambda)=\frac{1}{\pi J^2}\left(\lambda+\frac{s_J}{2}\right)^{3/2}\left(\frac{3s_J}{2}-\lambda\right)^{1/2}\mathbf{1}_{[-s_J/2,\,3s_J/2]}(\lambda). \tag*{(154)} σ ∗ ( λ ) = π J 2 1 ( λ + 2 s J ) 3/2 ( 2 3 s J − λ ) 1/2 1 [ − s J /2 , 3 s J /2 ] ( λ ) . (154) Thus the generic square-root left edge becomes a 3 / 2 3/2 3/2
The same quartic potential also determines the N N N λ = λ ∗ + δ λ \lambda=\lambda_*+\delta\lambda λ = λ ∗ + δ λ E = E ∗ + δ E E=E_*+\delta E E = E ∗ + δ E (153) gives
W E ( λ ) = μ F ∗ − δ E s J − 2 s J 2 ( δ E δ λ + δ λ 3 ) + O ( δ E 2 , δ E δ λ 2 , δ λ 4 ) . (155) W_E(\lambda)=\mu_F^*-\frac{\delta E}{s_J}-\frac{2}{s_J^2}\left(\delta E\delta\lambda+\delta\lambda^3\right)+O(\delta E^2,\delta E\delta\lambda^2,\delta\lambda^4). \tag*{(155)} W E ( λ ) = μ F ∗ − s J δ E − s J 2 2 ( δ E δ λ + δ λ 3 ) + O ( δ E 2 , δ E δ λ 2 , δ λ 4 ) . (155) The λ \lambda λ δ E \delta E δ E H N / N H_N/N H N / N N − 2 δ λ − 2 N^{-2}\delta\lambda^{-2} N − 2 δ λ − 2 δ λ 3 \delta\lambda^3 δ λ 3 δ E δ λ \delta E\delta\lambda δ E δ λ
δ λ = O ( N − 2 / 5 ) , δ E = O ( N − 4 / 5 ) . (156) \delta\lambda=O(N^{-2/5}),\qquad\delta E=O(N^{-4/5}). \tag*{(156)} δ λ = O ( N − 2/5 ) , δ E = O ( N − 4/5 ) . (156) Thus the N − 4 / 5 N^{-4/5} N − 4/5
Relation to the planar string equation
The orthogonal-polynomial analysis of the outgoing cubic matrix integral, developed in the next section, reaches exactly the same planar saddle in recurrence variables. At the endpoint n = N n=N n = N (185) , d 3 + E d + J 2 / 2 = 0 d^3+Ed+J^2/2=0 d 3 + E d + J 2 /2 = 0 (149) . The identification d = − λ d=-\lambda d = − λ (152) .
At the critical point this gives d ∗ = s J / 2 = − λ ∗ d_*=s_J/2=-\lambda_* d ∗ = s J /2 = − λ ∗ r ∗ = s J 2 / 4 r_*=s_J^2/4 r ∗ = s J 2 /4 N N N N N N N − 4 / 5 N^{-4/5} N − 4/5
String equations and critical regime
Complex orthogonal polynomials, string equations, and Toda identity
We give the details leading to Eq. (29) . For fixed N N N
⟨ f , g ⟩ − : = ∫ Γ − f ( λ ) g ( λ ) e − N V E ( λ ) d λ , V E ( λ ) = 2 J 2 ( λ 3 3 + E λ ) . (157) \langle f,g\rangle_- := \int_{\Gamma_-} f(\lambda)g(\lambda)e^{-NV_E(\lambda)}\,\mathrm{d}\lambda,\qquad V_E(\lambda)=\frac{2}{J^2}\left(\frac{\lambda^3}{3}+E\lambda\right). \tag*{(157)} ⟨ f , g ⟩ − := ∫ Γ − f ( λ ) g ( λ ) e − N V E ( λ ) d λ , V E ( λ ) = J 2 2 ( 3 λ 3 + E λ ) . (157) There is no complex conjugation in (S121); the product is therefore not positive, and the quantities below are in general complex. Let π n ( λ ) = λ n + ⋯ \pi_n(\lambda)=\lambda^n+\cdots π n ( λ ) = λ n + ⋯
⟨ π n , π m ⟩ − = h n δ n m . (158) \langle\pi_n,\pi_m\rangle_-=h_n\delta_{nm}. \tag*{(158)} ⟨ π n , π m ⟩ − = h n δ nm . (158) We suppress for the moment the N N N π n , N \pi_{n,N} π n , N β n , N \beta_{n,N} β n , N r n , N r_{n,N} r n , N N N N
It is useful to distinguish these polynomials from the polynomial eigenmodes p a p_a p a N N N p a p_a p a a = 1 , … , N a=1,\ldots,N a = 1 , … , N ℓ N 1 ∣ P N − 1 \ell_N^1\big|_{\mathcal{P}_{N-1}} ℓ N 1 P N − 1 π n \pi_n π n n = 0 , … , N − 1 n=0,\ldots,N-1 n = 0 , … , N − 1 1 , λ , … , λ N − 1 1,\lambda,\ldots,\lambda^{N-1} 1 , λ , … , λ N − 1
p a ( λ ) = ∑ n = 0 N − 1 A a n π n ( λ ) , A ∈ G L ( N , C ) , (159) p_a(\lambda)=\sum_{n=0}^{N-1}A_{an}\pi_n(\lambda),\qquad A\in GL(N,\mathbb{C}), \tag*{(159)} p a ( λ ) = n = 0 ∑ N − 1 A an π n ( λ ) , A ∈ G L ( N , C ) , (159) but they are adapted to different structures: the p a p_a p a π n \pi_n π n
For the following low-N N N
T N : = J 2 2 N , s N : = − E T N 2 / 3 , μ k ( N ) ( E ) : = ∫ Γ − λ k e − N V E ( λ ) d λ . (160) T_N:=\frac{J^2}{2N},\qquad s_N:=-\frac{E}{T_N^{2/3}},\qquad\mu_k^{(N)}(E):=\int_{\Gamma_-}\lambda^k e^{-NV_E(\lambda)}\,\mathrm{d}\lambda. \tag*{(160)} T N := 2 N J 2 , s N := − T N 2/3 E , μ k ( N ) ( E ) := ∫ Γ − λ k e − N V E ( λ ) d λ . (160) Using λ = − T N 1 / 3 w \lambda=-T_N^{1/3}w λ = − T N 1/3 w (24) gives the exact moment formula
μ k ( N ) ( E ) = i π T N ( k + 1 ) / 3 A − ( k ) ( s N ) . (161) \mu_k^{(N)}(E)=\mathrm{i}\pi T_N^{(k+1)/3}\mathcal{A}_{-}^{(k)}(s_N). \tag*{(161)} μ k ( N ) ( E ) = i π T N ( k + 1 ) /3 A − ( k ) ( s N ) . (161) Hence the first orthogonal polynomials are
N = 1 : π 0 ( 1 ) ( λ ) = 1 , N = 2 : π 0 ( 2 ) ( λ ) = 1 , π 1 ( 2 ) ( λ ) = λ − μ 1 ( 2 ) μ 0 ( 2 ) , N = 3 : π 0 ( 3 ) ( λ ) = 1 , π 1 ( 3 ) ( λ ) = λ − μ 1 ( 3 ) μ 0 ( 3 ) , π 2 ( 3 ) ( λ ) = λ 2 + μ 1 ( 3 ) μ 2 ( 3 ) − μ 0 ( 3 ) μ 3 ( 3 ) D 2 ( 3 ) λ + μ 1 ( 3 ) μ 3 ( 3 ) − ( μ 2 ( 3 ) ) 2 D 2 ( 3 ) , D 2 ( 3 ) : = μ 0 ( 3 ) μ 2 ( 3 ) − ( μ 1 ( 3 ) ) 2 . (162) \begin{aligned}
N=1:\quad&\pi_0^{(1)}(\lambda)=1,\\
N=2:\quad&\pi_0^{(2)}(\lambda)=1,\qquad\pi_1^{(2)}(\lambda)=\lambda-\frac{\mu_1^{(2)}}{\mu_0^{(2)}},\\
N=3:\quad&\pi_0^{(3)}(\lambda)=1,\qquad\pi_1^{(3)}(\lambda)=\lambda-\frac{\mu_1^{(3)}}{\mu_0^{(3)}},\\
&\pi_2^{(3)}(\lambda)=\lambda^2+\frac{\mu_1^{(3)}\mu_2^{(3)}-\mu_0^{(3)}\mu_3^{(3)}}{\mathfrak{D}_2^{(3)}}\lambda+\frac{\mu_1^{(3)}\mu_3^{(3)}-(\mu_2^{(3)})^2}{\mathfrak{D}_2^{(3)}},\\
&\mathfrak{D}_2^{(3)}:=\mu_0^{(3)}\mu_2^{(3)}-(\mu_1^{(3)})^2. \tag*{(162)}
\end{aligned} N = 1 : N = 2 : N = 3 : π 0 ( 1 ) ( λ ) = 1 , π 0 ( 2 ) ( λ ) = 1 , π 1 ( 2 ) ( λ ) = λ − μ 0 ( 2 ) μ 1 ( 2 ) , π 0 ( 3 ) ( λ ) = 1 , π 1 ( 3 ) ( λ ) = λ − μ 0 ( 3 ) μ 1 ( 3 ) , π 2 ( 3 ) ( λ ) = λ 2 + D 2 ( 3 ) μ 1 ( 3 ) μ 2 ( 3 ) − μ 0 ( 3 ) μ 3 ( 3 ) λ + D 2 ( 3 ) μ 1 ( 3 ) μ 3 ( 3 ) − ( μ 2 ( 3 ) ) 2 , D 2 ( 3 ) := μ 0 ( 3 ) μ 2 ( 3 ) − ( μ 1 ( 3 ) ) 2 . (162) In particular, even π 1 ( 2 ) \pi_1^{(2)} π 1 ( 2 ) π 1 ( 3 ) \pi_1^{(3)} π 1 ( 3 ) e − N V E e^{-NV_E} e − N V E N N N N N N
To see why multiplication by λ \lambda λ n + 1 n+1 n + 1 λ π n \lambda\pi_n λ π n π n + 1 \pi_{n+1} π n + 1 { π 0 , … , π n − 1 } \{\pi_0,\ldots,\pi_{n-1}\} { π 0 , … , π n − 1 } n − 1 n-1 n − 1 ⟨ π n , q ⟩ − = 0 \langle\pi_n,q\rangle_-=0 ⟨ π n , q ⟩ − = 0 deg q < n \deg q<n deg q < n k ≤ n − 2 k\le n-2 k ≤ n − 2 deg ( λ π k ) ≤ n − 1 \deg(\lambda\pi_k)\le n-1 deg ( λ π k ) ≤ n − 1 ⟨ λ π n , π k ⟩ − = ⟨ π n , λ π k ⟩ − = 0 \langle\lambda\pi_n,\pi_k\rangle_-=\langle\pi_n,\lambda\pi_k\rangle_-=0 ⟨ λ π n , π k ⟩ − = ⟨ π n , λ π k ⟩ − = 0
λ π n = π n + 1 + β n π n + r n π n − 1 , β n = ⟨ λ π n , π n ⟩ − h n , r n = h n h n − 1 . (163) \lambda\pi_n=\pi_{n+1}+\beta_n\pi_n+r_n\pi_{n-1},\qquad\beta_n=\frac{\langle\lambda\pi_n,\pi_n\rangle_-}{h_n},\qquad r_n=\frac{h_n}{h_{n-1}}. \tag*{(163)} λ π n = π n + 1 + β n π n + r n π n − 1 , β n = h n ⟨ λ π n , π n ⟩ − , r n = h n − 1 h n . (163) with r 0 = 0 r_0=0 r 0 = 0 π n − 1 \pi_{n-1} π n − 1 ⟨ λ π n , π n − 1 ⟩ − = r n h n − 1 \langle\lambda\pi_n,\pi_{n-1}\rangle_{-}=r_nh_{n-1} ⟨ λ π n , π n − 1 ⟩ − = r n h n − 1 ⟨ π n , λ π n − 1 ⟩ − = h n \langle\pi_n,\lambda\pi_{n-1}\rangle_{-}=h_n ⟨ π n , λ π n − 1 ⟩ − = h n π n − 1 \pi_{n-1} π n − 1 r n = h n / h n − 1 r_n=h_n/h_{n-1} r n = h n / h n − 1 N N N Z N ( − ) Z_N^{(-)} Z N ( − ) (25) . Since the π n \pi_n π n
V N ( λ ) = det 1 ≤ i , j ≤ N [ π j − 1 ( λ i ) ] . (164) V_N(\lambda)=\det_{1\leq i,j\leq N}\left[\pi_{j-1}(\lambda_i)\right]. \tag*{(164)} V N ( λ ) = 1 ≤ i , j ≤ N det [ π j − 1 ( λ i ) ] . (164) Applying Andréief’s identity to the two Vandermonde determinants gives
Z N ( − ) = det 0 ≤ j , k < N ⟨ π j , π k ⟩ − = ∏ n = 0 N − 1 h n . (165) Z_N^{(-)}=\det_{0\leq j,k<N}\langle\pi_j,\pi_k\rangle_{-}=\prod_{n=0}^{N-1}h_n. \tag*{(165)} Z N ( − ) = 0 ≤ j , k < N det ⟨ π j , π k ⟩ − = n = 0 ∏ N − 1 h n . (165) The norms themselves are ratios of successive Hankel minors. Using Eq. (161) , and defining τ 0 ( s ) : = 1 \tau_0(s):=1 τ 0 ( s ) := 1
τ n ( s ) : = det 0 ≤ j , k < n [ A ( j + k ) ( s ) ] , \tau_n(s):=\det_{0\leq j,k<n}\left[\mathcal{A}^{(j+k)}(s)\right], τ n ( s ) := 0 ≤ j , k < n det [ A ( j + k ) ( s ) ] , one obtains
h n = det 0 ≤ j , k ≤ n [ μ j + k ] det 0 ≤ j , k < n [ μ j + k ] = i π T ( 2 n + 1 ) / 3 τ n + 1 ( s ) τ n ( s ) , s = − E T 2 / 3 . (166) h_n=\frac{\det_{0\leq j,k\leq n}[\mu_{j+k}]}{\det_{0\leq j,k<n}[\mu_{j+k}]}=i\pi T^{(2n+1)/3}\frac{\tau_{n+1}(s)}{\tau_n(s)},\qquad s=-\frac{E}{T^{2/3}}. \tag*{(166)} h n = det 0 ≤ j , k < n [ μ j + k ] det 0 ≤ j , k ≤ n [ μ j + k ] = iπ T ( 2 n + 1 ) /3 τ n ( s ) τ n + 1 ( s ) , s = − T 2/3 E . (166) where here and below T = J 2 / ( 2 N ) T=J^2/(2N) T = J 2 / ( 2 N ) A : = A − ( s ) A:=\mathcal{A}_{-}(s) A := A − ( s )
h 0 = i π T 1 / 3 A , h_0=i\pi T^{1/3}A, h 0 = iπ T 1/3 A , h 1 = i π T s A 2 − ( A ′ ) 2 A , h_1=i\pi T\frac{sA^2-(A')^2}{A}, h 1 = iπ T A s A 2 − ( A ′ ) 2 , h 2 = i π T 5 / 3 − A 3 + 2 s A 2 A ′ − 2 ( A ′ ) 3 s A 2 − ( A ′ ) 2 . (167) h_2=i\pi T^{5/3}\frac{-A^3+2sA^2A'-2(A')^3}{sA^2-(A')^2}. \tag*{(167)} h 2 = iπ T 5/3 s A 2 − ( A ′ ) 2 − A 3 + 2 s A 2 A ′ − 2 ( A ′ ) 3 . (167) Consequently the complete product telescopes to
Z N ( − ) ( E ) = ∏ n = 0 N − 1 h n = ( i π ) N T N 2 / 3 τ N ( s ) . (168) Z_N^{(-)}(E)=\prod_{n=0}^{N-1}h_n=(i\pi)^N T^{N^2/3}\tau_N(s). \tag*{(168)} Z N ( − ) ( E ) = n = 0 ∏ N − 1 h n = ( iπ ) N T N 2 /3 τ N ( s ) . (168) Thus, with the contour orientation used here, the E E E (25) is C N = ( i π ) N C_N=(i\pi)^N C N = ( iπ ) N
The same orthogonal-polynomial construction defines an auxiliary finite-rank kernel on the cubic contour,
K N c u b ( λ , μ ) = e − N V E ( μ ) ∑ n = 0 N − 1 π n ( λ ) π n ( μ ) h n , λ , μ ∈ Γ − . (169) K_N^{\mathrm{cub}}(\lambda,\mu)=e^{-NV_E(\mu)}\sum_{n=0}^{N-1}\frac{\pi_n(\lambda)\pi_n(\mu)}{h_n},\qquad\lambda,\mu\in\Gamma_{-}. \tag*{(169)} K N cub ( λ , μ ) = e − N V E ( μ ) n = 0 ∑ N − 1 h n π n ( λ ) π n ( μ ) , λ , μ ∈ Γ − . (169) Equivalently, the Christoffel–Darboux identity gives
K N c u b ( λ , μ ) = e − N V E ( μ ) h N − 1 π N ( λ ) π N − 1 ( μ ) − π N − 1 ( λ ) π N ( μ ) λ − μ . (170) K_N^{\mathrm{cub}}(\lambda,\mu)=\frac{e^{-NV_E(\mu)}}{h_{N-1}}\frac{\pi_N(\lambda)\pi_{N-1}(\mu)-\pi_{N-1}(\lambda)\pi_N(\mu)}{\lambda-\mu}. \tag*{(170)} K N cub ( λ , μ ) = h N − 1 e − N V E ( μ ) λ − μ π N ( λ ) π N − 1 ( μ ) − π N − 1 ( λ ) π N ( μ ) . (170) Indeed, the normalized cubic-contour weight
1 N ! Z N ( − ) V N ( λ ) 2 ∏ j = 1 N e − N V E ( λ j ) d N λ , λ j ∈ Γ − , (171) \frac{1}{N!Z_N^{(-)}}V_N(\lambda)^2\prod_{j=1}^{N}e^{-NV_E(\lambda_j)}\,\mathrm{d}^N\lambda,\qquad\lambda_j\in\Gamma_{-}, \tag*{(171)} N ! Z N ( − ) 1 V N ( λ ) 2 j = 1 ∏ N e − N V E ( λ j ) d N λ , λ j ∈ Γ − , (171) has algebraic k k k
R k , N c u b ( λ 1 , … , λ k ) = det 1 ≤ i , j ≤ k [ K N c u b ( λ i , λ j ) ] . (172) R_{k,N}^{\mathrm{cub}}(\lambda_1,\ldots,\lambda_k)=\det_{1\leq i,j\leq k}\left[K_N^{\mathrm{cub}}(\lambda_i,\lambda_j)\right]. \tag*{(172)} R k , N cub ( λ 1 , … , λ k ) = 1 ≤ i , j ≤ k det [ K N cub ( λ i , λ j ) ] . (172) Since Γ − \Gamma_{-} Γ − e − N V E ( λ ) d λ e^{-NV_E(\lambda)}\,\mathrm{d}\lambda e − N V E ( λ ) d λ (171) is in general a normalized complex weight rather than a probability distribution. We therefore refer to it as a determinantal contour ensemble , not as a determinantal point process; its determinantal identities are algebraic consequences of Andréief’s formula, but its correlation functions need not be real or non-negative.
The contour kernel (S133) is therefore distinct from the physical stationary Riccati kernel (S67). The latter acts on R \mathbb{R} R ( p a , v a ) (p_a,v_a) ( p a , v a ) K N c u b K_N^{\mathrm{cub}} K N cub Γ − \Gamma_- Γ − ( π n , e − N V E π n / h n ) (\pi_n,e^{-NV_E}\pi_n/h_n) ( π n , e − N V E π n / h n ) P N − 1 \mathcal{P}_{N-1} P N − 1 (159) , while their right sectors are different. The connection between them is the Fourier–Airy transformation derived in End Matter: the Fourier transforms of the physical modes v a v_a v a A − ( n ) \mathcal{A}_-^{(n)} A − ( n ) Z N ( − ) \mathcal{Z}_N^{(-)} Z N ( − )
The string equations follow from two integrations by parts. Since the weight decays at both ends of Γ − \Gamma_- Γ −
0 = ∫ Γ − ∂ λ [ π n ( λ ) 2 e − N V E ( λ ) ] d λ . (173) 0 = \int_{\Gamma_-} \partial_\lambda\left[\pi_n(\lambda)^2 e^{-NV_E(\lambda)}\right]\,\mathrm{d}\lambda. \tag*{(173)} 0 = ∫ Γ − ∂ λ [ π n ( λ ) 2 e − N V E ( λ ) ] d λ . (173) Expanding the derivative gives
0 = 2 ⟨ π n ′ , π n ⟩ − − N ⟨ V E ′ π n , π n ⟩ − . 0 = 2\langle\pi_n',\pi_n\rangle_- - N\langle V_E'\pi_n,\pi_n\rangle_-. 0 = 2 ⟨ π n ′ , π n ⟩ − − N ⟨ V E ′ π n , π n ⟩ − . Because π n ′ \pi_n' π n ′ n − 1 n-1 n − 1 ⟨ π n ′ , π n ⟩ − = 0 \langle\pi_n',\pi_n\rangle_- = 0 ⟨ π n ′ , π n ⟩ − = 0 ⟨ V E ′ π n , π n ⟩ − = 0 \langle V_E'\pi_n,\pi_n\rangle_- = 0 ⟨ V E ′ π n , π n ⟩ − = 0 V E ′ = 2 ( λ 2 + E ) / J 2 V_E' = 2(\lambda^2+E)/J^2 V E ′ = 2 ( λ 2 + E ) / J 2
λ 2 π n = λ π n + 1 + β n λ π n + r n λ π n − 1 . \lambda^2\pi_n = \lambda\pi_{n+1}+\beta_n\lambda\pi_n+r_n\lambda\pi_{n-1}. λ 2 π n = λ π n + 1 + β n λ π n + r n λ π n − 1 . The coefficient of π n \pi_n π n r n + 1 r_{n+1} r n + 1 β n 2 \beta_n^2 β n 2 r n r_n r n π n \pi_n π n
⟨ λ 2 π n , π n ⟩ − = ( β n 2 + r n + r n + 1 ) h n . (174) \langle\lambda^2\pi_n,\pi_n\rangle_- = (\beta_n^2+r_n+r_{n+1})h_n. \tag*{(174)} ⟨ λ 2 π n , π n ⟩ − = ( β n 2 + r n + r n + 1 ) h n . (174) leading to the first string equation in Eq. (29) . Similarly,
0 = ∫ Γ − ∂ λ [ π n ( λ ) π n − 1 ( λ ) e − N V E ( λ ) ] d λ . (175) 0 = \int_{\Gamma_-} \partial_\lambda\left[\pi_n(\lambda)\pi_{n-1}(\lambda)e^{-NV_E(\lambda)}\right]\,\mathrm{d}\lambda. \tag*{(175)} 0 = ∫ Γ − ∂ λ [ π n ( λ ) π n − 1 ( λ ) e − N V E ( λ ) ] d λ . (175) Expanding Eq. (175) gives
0 = ⟨ π n ′ , π n − 1 ⟩ − + ⟨ π n , π n − 1 ′ ⟩ − − N ⟨ V E ′ π n , π n − 1 ⟩ − . 0 = \langle\pi_n',\pi_{n-1}\rangle_-+\langle\pi_n,\pi_{n-1}'\rangle_- - N\langle V_E'\pi_n,\pi_{n-1}\rangle_-. 0 = ⟨ π n ′ , π n − 1 ⟩ − + ⟨ π n , π n − 1 ′ ⟩ − − N ⟨ V E ′ π n , π n − 1 ⟩ − . Since π n − 1 ′ \pi_{n-1}' π n − 1 ′ n − 2 n-2 n − 2 ⟨ π n , π n − 1 ′ ⟩ − = 0 \langle\pi_n,\pi_{n-1}'\rangle_- = 0 ⟨ π n , π n − 1 ′ ⟩ − = 0 π n ′ = n π n − 1 + ( lower-degree terms ) \pi_n'=n\pi_{n-1}+(\text{lower-degree terms}) π n ′ = n π n − 1 + ( lower-degree terms ) ⟨ π n ′ , π n − 1 ⟩ − = n h n − 1 \langle\pi_n',\pi_{n-1}\rangle_- = nh_{n-1} ⟨ π n ′ , π n − 1 ⟩ − = n h n − 1 V E ′ = 2 ( λ 2 + E ) / J 2 V_E'=2(\lambda^2+E)/J^2 V E ′ = 2 ( λ 2 + E ) / J 2 E E E
⟨ λ 2 π n , π n − 1 ⟩ − = n J 2 2 N h n − 1 . \langle\lambda^2\pi_n,\pi_{n-1}\rangle_- = \frac{nJ^2}{2N}h_{n-1}. ⟨ λ 2 π n , π n − 1 ⟩ − = 2 N n J 2 h n − 1 . On the other hand, multiplying λ π n = π n + 1 + β n π n + r n π n − 1 \lambda\pi_n=\pi_{n+1}+\beta_n\pi_n+r_n\pi_{n-1} λ π n = π n + 1 + β n π n + r n π n − 1 λ \lambda λ π n − 1 \pi_{n-1} π n − 1
⟨ λ 2 π n , π n − 1 ⟩ − = ⟨ λ π n + 1 , π n − 1 ⟩ − + β n ⟨ λ π n , π n − 1 ⟩ − + r n ⟨ λ π n − 1 , π n − 1 ⟩ − . \begin{aligned}
\langle\lambda^2\pi_n,\pi_{n-1}\rangle_- &=
\langle\lambda\pi_{n+1},\pi_{n-1}\rangle_-+\beta_n\langle\lambda\pi_n,\pi_{n-1}\rangle_- \\
&\quad+r_n\langle\lambda\pi_{n-1},\pi_{n-1}\rangle_-.
\end{aligned} ⟨ λ 2 π n , π n − 1 ⟩ − = ⟨ λ π n + 1 , π n − 1 ⟩ − + β n ⟨ λ π n , π n − 1 ⟩ − + r n ⟨ λ π n − 1 , π n − 1 ⟩ − . The first term vanishes by orthogonality, while the recurrence gives ⟨ λ π n , π n − 1 ⟩ − = r n h n − 1 \langle\lambda\pi_n,\pi_{n-1}\rangle_- = r_nh_{n-1} ⟨ λ π n , π n − 1 ⟩ − = r n h n − 1 ⟨ λ π n − 1 , π n − 1 ⟩ − = β n − 1 h n − 1 \langle\lambda\pi_{n-1},\pi_{n-1}\rangle_- = \beta_{n-1}h_{n-1} ⟨ λ π n − 1 , π n − 1 ⟩ − = β n − 1 h n − 1
⟨ λ 2 π n , π n − 1 ⟩ − = r n ( β n + β n − 1 ) h n − 1 , \langle\lambda^2\pi_n,\pi_{n-1}\rangle_- = r_n(\beta_n+\beta_{n-1})h_{n-1}, ⟨ λ 2 π n , π n − 1 ⟩ − = r n ( β n + β n − 1 ) h n − 1 , and comparison with the previous expression yields the second string equation in Eq. (29) . Thus the string equations are exact finite-N N N Γ − \Gamma_- Γ −
We finally record the Toda identity used below. Introduce θ = − 2 N E / J 2 \theta=-2NE/J^2 θ = − 2 N E / J 2 E E E e θ λ e^{\theta\lambda} e θ λ π n \pi_n π n θ \theta θ ∂ θ π n \partial_\theta\pi_n ∂ θ π n n − 1 n-1 n − 1 h n = ⟨ π n , π n ⟩ − h_n=\langle\pi_n,\pi_n\rangle_- h n = ⟨ π n , π n ⟩ −
∂ θ h n = 2 ⟨ ∂ θ π n , π n ⟩ − + ⟨ π n , π n ⟩ − . \partial_\theta h_n=2\langle\partial_\theta\pi_n,\pi_n\rangle_-+\langle\pi_n,\pi_n\rangle_-. ∂ θ h n = 2 ⟨ ∂ θ π n , π n ⟩ − + ⟨ π n , π n ⟩ − . The first term vanishes by orthogonality, while the recurrence relation gives ⟨ λ π n , π n ⟩ − = β n h n \langle\lambda\pi_n,\pi_n\rangle_-=\beta_nh_n ⟨ λ π n , π n ⟩ − = β n h n
∂ θ log h n = β n . (176) \partial_\theta\log h_n=\beta_n. \tag*{(176)} ∂ θ log h n = β n . (176) To determine ∂ θ π n \partial_\theta\pi_n ∂ θ π n ⟨ π n , π k ⟩ − = 0 \langle\pi_n,\pi_k\rangle_-=0 ⟨ π n , π k ⟩ − = 0 k < n k<n k < n
0 = ⟨ ∂ θ π n , π k ⟩ − + ⟨ π n , ∂ θ π k ⟩ − + ⟨ λ π n , π k ⟩ − . 0=\langle\partial_\theta\pi_n,\pi_k\rangle_-+\langle\pi_n,\partial_\theta\pi_k\rangle_-+\langle\lambda\pi_n,\pi_k\rangle_-. 0 = ⟨ ∂ θ π n , π k ⟩ − + ⟨ π n , ∂ θ π k ⟩ − + ⟨ λ π n , π k ⟩ − . The second term vanishes since ∂ θ π k \partial_\theta\pi_k ∂ θ π k n n n k ≤ n − 2 k\le n-2 k ≤ n − 2 k = n − 1 k=n-1 k = n − 1 r n h n − 1 = h n r_nh_{n-1}=h_n r n h n − 1 = h n ∂ θ π n \partial_\theta\pi_n ∂ θ π n n − 1 n-1 n − 1
∂ θ π n = − r n π n − 1 . \partial_\theta\pi_n=-r_n\pi_{n-1}. ∂ θ π n = − r n π n − 1 . Writing π n ( λ ) = λ n + c n λ n − 1 + ⋯ \pi_n(\lambda)=\lambda^n+c_n\lambda^{n-1}+\cdots π n ( λ ) = λ n + c n λ n − 1 + ⋯ β n = c n − c n + 1 \beta_n=c_n-c_{n+1} β n = c n − c n + 1 ∂ θ c n = − r n \partial_\theta c_n=-r_n ∂ θ c n = − r n
∂ θ β n = r n + 1 − r n . (177) \partial_\theta\beta_n=r_{n+1}-r_n. \tag*{(177)} ∂ θ β n = r n + 1 − r n . (177) Using Z N ( − ) = ∏ n = 0 N − 1 h n Z_N^{(-)}=\prod_{n=0}^{N-1}h_n Z N ( − ) = ∏ n = 0 N − 1 h n (176) gives
∂ θ log Z N ( − ) = ∑ n = 0 N − 1 β n . (178) \partial_\theta\log Z_N^{(-)}=\sum_{n=0}^{N-1}\beta_n. \tag*{(178)} ∂ θ log Z N ( − ) = n = 0 ∑ N − 1 β n . (178) Differentiating once more and using Eq. (177) ,
∂ θ 2 log Z N ( − ) = ∑ n = 0 N − 1 ( r n + 1 − r n ) = r N ⇒ ∂ E 2 log Z N ( − ) = ( 2 N J 2 ) 2 r N . (179) \partial_\theta^2\log Z_N^{(-)}=\sum_{n=0}^{N-1}(r_{n+1}-r_n)=r_N
\quad\Rightarrow\quad
\partial_E^2\log Z_N^{(-)}=\left(\frac{2N}{J^2}\right)^2r_N. \tag*{(179)} ∂ θ 2 log Z N ( − ) = n = 0 ∑ N − 1 ( r n + 1 − r n ) = r N ⇒ ∂ E 2 log Z N ( − ) = ( J 2 2 N ) 2 r N . (179) since the sum telescopes, and since r 0 = 0 r_0=0 r 0 = 0 θ = − 2 N E / J 2 \theta=-2NE/J^2 θ = − 2 N E / J 2
Planar N = ∞ N=\infty N = ∞
We now restore explicitly the N N N β n , N \beta_{n,N} β n , N r n , N r_{n,N} r n , N (29) hold at every finite N N N N → ∞ N\to\infty N → ∞
t = n N (180) t=\frac{n}{N} \tag*{(180)} t = N n (180) fixed. Away from a critical point, the recurrence coefficients are assumed to approach smooth functions of t t t
β n , N ⟶ d ( t ) , r n , N ⟶ r ( t ) , N → ∞ . (181) \beta_{n,N}\longrightarrow d(t),\qquad r_{n,N}\longrightarrow r(t),\qquad N\to\infty. \tag*{(181)} β n , N ⟶ d ( t ) , r n , N ⟶ r ( t ) , N → ∞. (181) This is the N = ∞ N=\infty N = ∞ n n n t t t 1 / N 1/N 1/ N
r n + 1 , N = r ( t ) + O ( N − 1 ) , β n − 1 , N = d ( t ) + O ( N − 1 ) . (182) r_{n+1,N}=r(t)+O(N^{-1}),\qquad\beta_{n-1,N}=d(t)+O(N^{-1}). \tag*{(182)} r n + 1 , N = r ( t ) + O ( N − 1 ) , β n − 1 , N = d ( t ) + O ( N − 1 ) . (182) Substituting these leading expressions into the exact string equations gives
d ( t ) 2 + 2 r ( t ) + E = 0 , 2 r ( t ) d ( t ) = t J 2 2 . (183) d(t)^2+2r(t)+E=0,\qquad2r(t)d(t)=\frac{tJ^2}{2}. \tag*{(183)} d ( t ) 2 + 2 r ( t ) + E = 0 , 2 r ( t ) d ( t ) = 2 t J 2 . (183) Eliminating r ( t ) r(t) r ( t )
F ( d ; t , E ) : = d 3 + E d + t J 2 2 = 0. (184) F(d;t,E):=d^3+Ed+\frac{tJ^2}{2}=0. \tag*{(184)} F ( d ; t , E ) := d 3 + E d + 2 t J 2 = 0. (184) The physical N × N N\times N N × N n = N n=N n = N t = 1 t=1 t = 1 r N , N r_{N,N} r N , N
d 3 + E d + J 2 2 = 0. (185) d^3+Ed+\frac{J^2}{2}=0. \tag*{(185)} d 3 + E d + 2 J 2 = 0. (185) This is precisely the algebraic equation governing the large-N N N § 0.9 C, Eq. (185) is identical to the cubic equation for d d d (149) . The critical point occurs when two roots of the cubic merge,
F ( d ∗ ; 1 , E ∗ ) = 0 , ∂ d F ( d ∗ ; 1 , E ∗ ) = 3 d ∗ 2 + E ∗ = 0. (186) F(d_*;1,E^*)=0,\qquad\partial_d F(d_*;1,E^*)=3d_*^2+E^*=0. \tag*{(186)} F ( d ∗ ; 1 , E ∗ ) = 0 , ∂ d F ( d ∗ ; 1 , E ∗ ) = 3 d ∗ 2 + E ∗ = 0. (186) With s J = ( 2 J 2 ) 1 / 3 s_J=(2J^2)^{1/3} s J = ( 2 J 2 ) 1/3
d ∗ = s J 2 , r ∗ = s J 2 4 , E ∗ = − 3 s J 2 4 . (187) d_*=\frac{s_J}{2},\qquad r_*=\frac{s_J^2}{4},\qquad E^*=-\frac{3s_J^2}{4}. \tag*{(187)} d ∗ = 2 s J , r ∗ = 4 s J 2 , E ∗ = − 4 3 s J 2 . (187) This is the same large-N N N [28 , 30 ] , and corresponds to the confined-to-flowing transition of the nonconfining Dyson gas found by Allez and Dumaz [34 ] .
It is useful to distinguish this strict N = ∞ N=\infty N = ∞ N N N t t t
r n + 1 , N = r ( t + 1 N ) = r ( t ) + 1 N r ′ ( t ) + ⋯ , (188) r_{n+1,N}=r\left(t+\frac{1}{N}\right)=r(t)+\frac{1}{N}r'(t)+\cdots, \tag*{(188)} r n + 1 , N = r ( t + N 1 ) = r ( t ) + N 1 r ′ ( t ) + ⋯ , (188) and similarly for β n ± 1 , N \beta_{n\pm1,N} β n ± 1 , N N N N
At the edge this regular expansion becomes singular. Indeed, differentiating Eq. (184) with respect to t t t
( 3 d 2 + E ) ∂ t d + J 2 2 = 0 , ∂ t d = − J 2 / 2 3 d 2 + E . (189) (3d^2+E)\,\partial_t d+\frac{J^2}{2}=0,\qquad\partial_t d=-\frac{J^2/2}{3d^2+E}. \tag*{(189)} ( 3 d 2 + E ) ∂ t d + 2 J 2 = 0 , ∂ t d = − 3 d 2 + E J 2 /2 . (189) The denominator vanishes at ( d ∗ , E ∗ ) (d_*,E^*) ( d ∗ , E ∗ ) t t t N N N
The same singularity is visible directly in the dependence on E E E d = d ∗ + δ d d=d_*+\delta d d = d ∗ + δ d E = E ∗ + δ E E=E^*+\delta E E = E ∗ + δ E (185) , the term linear in δ d \delta d δ d
d ∗ δ E + 3 d ∗ ( δ d ) 2 + ⋯ = 0 , i.e. E − E ∗ ≃ − 3 ( d − d ∗ ) 2 . (190) d_*\,\delta E+3d_*(\delta d)^2+\cdots=0,\qquad\text{i.e.}\quad E-E^*\simeq-3(d-d_*)^2. \tag*{(190)} d ∗ δ E + 3 d ∗ ( δ d ) 2 + ⋯ = 0 , i.e. E − E ∗ ≃ − 3 ( d − d ∗ ) 2 . (190) Thus the two planar branches separate with a square-root singularity, d − d ∗ ∼ ( E ∗ − E ) 1 / 2 d-d_*\sim(E^*-E)^{1/2} d − d ∗ ∼ ( E ∗ − E ) 1/2 1 / N 1/N 1/ N n → n ± 1 n\to n\pm1 n → n ± 1 n − N n-N n − N β n , N \beta_{n,N} β n , N r n , N r_{n,N} r n , N E − E ∗ E-E^* E − E ∗
Double scaling and the Painlevé-I equation
We now resolve the breakdown of the planar approximation at the double root. Let ℓ \ell ℓ A A A β n , N \beta_{n,N} β n , N r n , N r_{n,N} r n , N A 2 A^2 A 2 n n n A / ℓ 2 A/\ell^2 A / ℓ 2 ℓ \ell ℓ n = N n=N n = N t = n / N t=n/N t = n / N O ( ℓ / N ) O(\ell/N) O ( ℓ / N )
A 2 ∼ A ℓ 2 ∼ ℓ N . (191) A^2\sim\frac{A}{\ell^2}\sim\frac{\ell}{N}. \tag*{(191)} A 2 ∼ ℓ 2 A ∼ N ℓ . (191) The first equality gives A ∼ ℓ − 2 A\sim\ell^{-2} A ∼ ℓ − 2 ℓ 5 ∼ N \ell^5\sim N ℓ 5 ∼ N
ℓ ∼ N 1 / 5 , A ∼ N − 2 / 5 . (192) \ell\sim N^{1/5},\qquad A\sim N^{-2/5}. \tag*{(192)} ℓ ∼ N 1/5 , A ∼ N − 2/5 . (192) Since near the double root E − E ∗ = O ( A 2 ) E-E^*=O(A^2) E − E ∗ = O ( A 2 )
E − E ∗ = O ( N − 4 / 5 ) . (193) E-E^*=O(N^{-4/5}). \tag*{(193)} E − E ∗ = O ( N − 4/5 ) . (193) This explains the three scales quoted in the main text.
It is convenient to introduce
δ = N − 1 / 5 , ξ n = ( n − N ) δ , E = E ∗ + s J 2 δ 4 X . (194) \delta= N^{-1/5}, \qquad\xi_n = (n-N)\delta, \qquad E = E^* + s_J^2\delta^4 X. \tag*{(194)} δ = N − 1/5 , ξ n = ( n − N ) δ , E = E ∗ + s J 2 δ 4 X . (194) Then
n N = 1 + δ 4 ξ n , n → n ± 1 ⟺ ξ n → ξ n ± δ . (195) \frac{n}{N}=1+\delta^4\xi_n,\qquad n\to n\pm1\Longleftrightarrow\xi_n\to\xi_n\pm\delta. \tag*{(195)} N n = 1 + δ 4 ξ n , n → n ± 1 ⟺ ξ n → ξ n ± δ . (195) Thus the O ( N 1 / 5 ) O(N^{1/5}) O ( N 1/5 ) n = N n=N n = N ξ = O ( 1 ) \xi=O(1) ξ = O ( 1 ) β n , N \beta_{n,N} β n , N n n n r n , N r_{n,N} r n , N π n − 1 \pi_{n-1} π n − 1 n − 1 2 n-\frac{1}{2} n − 2 1
β n , N = s J [ 1 2 + δ 2 b ( ξ n ) + δ 4 b 1 ( ξ n ) + ⋯ ] , r n , N = s J 2 [ 1 4 + δ 2 a ( ξ n − δ 2 ) + δ 4 a 1 ( ξ n − δ 2 ) + ⋯ ] . (196) \begin{aligned}
\beta_{n,N}&=s_J\left[\frac{1}{2}+\delta^2b(\xi_n)+\delta^4b_1(\xi_n)+\cdots\right],\\
r_{n,N}&=s_J^2\left[\frac{1}{4}+\delta^2a\left(\xi_n-\frac{\delta}{2}\right)+\delta^4a_1\left(\xi_n-\frac{\delta}{2}\right)+\cdots\right]. \tag*{(196)}
\end{aligned} β n , N r n , N = s J [ 2 1 + δ 2 b ( ξ n ) + δ 4 b 1 ( ξ n ) + ⋯ ] , = s J 2 [ 4 1 + δ 2 a ( ξ n − 2 δ ) + δ 4 a 1 ( ξ n − 2 δ ) + ⋯ ] . (196) The midpoint convention makes the combinations entering the string equations symmetric. For example,
a ( ξ − δ 2 ) + a ( ξ + δ 2 ) = 2 a ( ξ ) + δ 2 4 a ′ ′ ( ξ ) + O ( δ 4 ) . (197) a\left(\xi-\frac{\delta}{2}\right)+a\left(\xi+\frac{\delta}{2}\right)=2a(\xi)+\frac{\delta^2}{4}a''(\xi)+O(\delta^4). \tag*{(197)} a ( ξ − 2 δ ) + a ( ξ + 2 δ ) = 2 a ( ξ ) + 4 δ 2 a ′′ ( ξ ) + O ( δ 4 ) . (197) Consider first β n , N 2 + r n , N + r n + 1 , N + E = 0 \beta_{n,N}^2+r_{n,N}+r_{n+1,N}+E=0 β n , N 2 + r n , N + r n + 1 , N + E = 0 d ∗ = s J / 2 d_*=s_J/2 d ∗ = s J /2 r ∗ = s J 2 / 4 r_*=s_J^2/4 r ∗ = s J 2 /4 E ∗ = − 3 s J 2 / 4 E^*=-3s_J^2/4 E ∗ = − 3 s J 2 /4 δ 2 \delta^2 δ 2
b + 2 a = 0. (198) b+2a=0. \tag*{(198)} b + 2 a = 0. (198) At the next order, δ 4 \delta^4 δ 4
b 2 + b 1 + 2 a 1 + 1 4 a ′ ′ + X = 0. (199) b^2+b_1+2a_1+\frac{1}{4}a''+X=0. \tag*{(199)} b 2 + b 1 + 2 a 1 + 4 1 a ′′ + X = 0. (199) We next expand the second string equation,
r n , N ( β n , N + β n − 1 , N ) = n J 2 2 N . (200) r_{n,N}(\beta_{n,N}+\beta_{n-1,N})=\frac{nJ^2}{2N}. \tag*{(200)} r n , N ( β n , N + β n − 1 , N ) = 2 N n J 2 . (200) It is natural here to expand about the bond n − 1 2 n-\frac{1}{2} n − 2 1
b ( ξ + δ 2 ) + b ( ξ − δ 2 ) = 2 b ( ξ ) + δ 2 4 b ′ ′ ( ξ ) + O ( δ 4 ) . (201) b\left(\xi+\frac{\delta}{2}\right)+b\left(\xi-\frac{\delta}{2}\right)=2b(\xi)+\frac{\delta^2}{4}b''(\xi)+O(\delta^4). \tag*{(201)} b ( ξ + 2 δ ) + b ( ξ − 2 δ ) = 2 b ( ξ ) + 4 δ 2 b ′′ ( ξ ) + O ( δ 4 ) . (201) Using s J 3 = 2 J 2 s_J^3=2J^2 s J 3 = 2 J 2
n J 2 2 N = s J 3 4 ( 1 + δ 4 ξ + ⋯ ) . (202) \frac{nJ^2}{2N}=\frac{s_J^3}{4}\left(1+\delta^4\xi+\cdots\right). \tag*{(202)} 2 N n J 2 = 4 s J 3 ( 1 + δ 4 ξ + ⋯ ) . (202) At order δ 2 \delta^2 δ 2
a + b 2 = 0 , (203) a+\frac{b}{2}=0, \tag*{(203)} a + 2 b = 0 , (203) which is identical to Eq. (198) . At order δ 4 \delta^4 δ 4
a 1 + b 1 2 + 1 16 b ′ ′ + 2 a b = ξ 4 . (204) a_1+\frac{b_1}{2}+\frac{1}{16}b''+2ab=\frac{\xi}{4}. \tag*{(204)} a 1 + 2 b 1 + 16 1 b ′′ + 2 ab = 4 ξ . (204) The two O ( δ 4 ) O(\delta^4) O ( δ 4 ) a 1 a_1 a 1 b 1 b_1 b 1 (198) , b = − 2 a b=-2a b = − 2 a
b 1 + 2 a 1 = − b 2 − 1 4 a ′ ′ − X (205) b_1+2a_1=-b^2-\frac{1}{4}a''-X \tag*{(205)} b 1 + 2 a 1 = − b 2 − 4 1 a ′′ − X (205) and
b 1 + 2 a 1 = ξ 2 − 1 8 b ′ ′ − 4 a b . (206) b_1+2a_1=\frac{\xi}{2}-\frac{1}{8}b''-4ab. \tag*{(206)} b 1 + 2 a 1 = 2 ξ − 8 1 b ′′ − 4 ab . (206) Equating these two expressions and using b = − 2 a b=-2a b = − 2 a
a ′ ′ ( ξ ) + 24 a ( ξ ) 2 + ξ + 2 X = 0. (207) a''(\xi)+24a(\xi)^2+\xi+2X=0. \tag*{(207)} a ′′ ( ξ ) + 24 a ( ξ ) 2 + ξ + 2 X = 0. (207) To bring it to canonical form, define
u ( ξ ) = − 4 a ( ξ ) . (208) u(\xi)=-4a(\xi). \tag*{(208)} u ( ξ ) = − 4 a ( ξ ) . (208) Then
u ′ ′ = 6 u 2 + 4 ( ξ + 2 X ) . (209) u''=6u^2+4(\xi+2X). \tag*{(209)} u ′′ = 6 u 2 + 4 ( ξ + 2 X ) . (209) Finally introduce
t = 4 1 / 5 ( ξ + 2 X ) , u ( ξ ) = 4 2 / 5 y ( t ) . (210) t=4^{1/5}(\xi+2X),\qquad u(\xi)=4^{2/5}y(t). \tag*{(210)} t = 4 1/5 ( ξ + 2 X ) , u ( ξ ) = 4 2/5 y ( t ) . (210) Equation (S173) becomes
y ′ ′ ( t ) = 6 y ( t ) 2 + t , (211) y''(t)=6y(t)^2+t, \tag*{(211)} y ′′ ( t ) = 6 y ( t ) 2 + t , (211) which is the canonical Painlevé-I equation.
Using b = − 2 a b=-2a b = − 2 a
β n , N = s J 2 + s J 2 1 / 5 N 2 / 5 y ( t ) + ⋯ , r n , N = s J 2 4 − s J 2 2 6 / 5 N 2 / 5 y ( t ) + ⋯ . (212) \begin{aligned}
\beta_{n,N}&=\frac{s_J}{2}+\frac{s_J}{2^{1/5}N^{2/5}}y(t)+\cdots,\\
r_{n,N}&=\frac{s_J^2}{4}-\frac{s_J^2}{2^{6/5}N^{2/5}}y(t)+\cdots. \tag*{(212)}
\end{aligned} β n , N r n , N = 2 s J + 2 1/5 N 2/5 s J y ( t ) + ⋯ , = 4 s J 2 − 2 6/5 N 2/5 s J 2 y ( t ) + ⋯ . (212) with
t = 2 7 / 5 X + 4 1 / 5 ( n − N ) N − 1 / 5 . (213) t=2^{7/5}X+4^{1/5}(n-N)N^{-1/5}. \tag*{(213)} t = 2 7/5 X + 4 1/5 ( n − N ) N − 1/5 . (213) This is the double-scaling ansatz quoted in the End Matter. At this stage the discrete string equations determine the Painlevé-I equation ; the choice of the particular solution requires the additional information carried by the contour Γ − \Gamma_- Γ −
From the recurrence coefficient to the density. The Toda identity (S143) directly relates the endpoint recurrence coefficient to the spectral observables. Since
Ξ N ( E ) = − T ∂ E log Z N ( − ) ( E ) , T = J 2 2 N , \Xi_N(E)=-T\partial_E\log Z_N^{(-)}(E),\qquad T=\frac{J^2}{2N}, Ξ N ( E ) = − T ∂ E log Z N ( − ) ( E ) , T = 2 N J 2 , one finds
∂ E Ξ N ( E ) = − T ∂ E 2 log Z N ( − ) = − r N , N T . (214) \partial_E\Xi_N(E)=-T\partial_E^2\log Z_N^{(-)}=-\frac{r_{N,N}}{T}. \tag*{(214)} ∂ E Ξ N ( E ) = − T ∂ E 2 log Z N ( − ) = − T r N , N . (214) Writing Ξ N = Γ N − i π ν N \Xi_N=\Gamma_N-i\pi\nu_N Ξ N = Γ N − iπ ν N ρ N = ∂ E ν N \rho_N=\partial_E\nu_N ρ N = ∂ E ν N
ρ N ( E ) = 1 π T Im r N , N . (215) \rho_N(E)=\frac{1}{\pi T}\operatorname{Im}r_{N,N}. \tag*{(215)} ρ N ( E ) = π T 1 Im r N , N . (215) Thus only the imaginary part of the critical correction to r N , N r_{N,N} r N , N r ∗ = s J 2 / 4 r_* =s_J^2/4 r ∗ = s J 2 /4
At n = N n=N n = N ξ = 0 \xi=0 ξ = 0 t = 2 7 / 5 X t=2^{7/5}X t = 2 7/5 X
r N , N = s J 2 4 − s J 2 2 6 / 5 N 2 / 5 y ( 2 7 / 5 X ) + ⋯ . (216) r_{N,N}=\frac{s_J^2}{4}-\frac{s_J^2}{2^{6/5}N^{2/5}}y(2^{7/5}X)+\cdots. \tag*{(216)} r N , N = 4 s J 2 − 2 6/5 N 2/5 s J 2 y ( 2 7/5 X ) + ⋯ . (216) Since s J 3 = 2 J 2 s_J^3 = 2J^2 s J 3 = 2 J 2 T = s J 3 / ( 4 N ) T = s_J^3/(4N) T = s J 3 / ( 4 N ) (215) therefore yields
ρ N ( E ) ∼ − 2 4 / 5 N 3 / 5 π s J Im y ( 2 7 / 5 X ) . (217) \rho_N(E) \sim-\frac{2^{4/5}N^{3/5}}{\pi s_J}\operatorname{Im} y(2^{7/5}X). \tag*{(217)} ρ N ( E ) ∼ − π s J 2 4/5 N 3/5 Im y ( 2 7/5 X ) . (217) Defining
Φ ′ ( X ) : = − 2 4 / 5 π Im y ( 2 7 / 5 X ) , (218) \Phi'(X) := -\frac{2^{4/5}}{\pi}\operatorname{Im} y(2^{7/5}X), \tag*{(218)} Φ ′ ( X ) := − π 2 4/5 Im y ( 2 7/5 X ) , (218) one recovers
ρ N ( E ) ∼ N 3 / 5 s J Φ ′ ( X ) . (219) \rho_N(E) \sim\frac{N^{3/5}}{s_J}\Phi'(X). \tag*{(219)} ρ N ( E ) ∼ s J N 3/5 Φ ′ ( X ) . (219) Finally, since E = E ∗ + s J 2 N − 4 / 5 X E = E^* + s_J^2N^{-4/5}X E = E ∗ + s J 2 N − 4/5 X ρ N = ∂ E ν N \rho_N = \partial_E\nu_N ρ N = ∂ E ν N
ν N ( E ) = s J N − 1 / 5 Φ ( X ) , (220) \nu_N(E) = s_JN^{-1/5}\Phi(X), \tag*{(220)} ν N ( E ) = s J N − 1/5 Φ ( X ) , (220) up to an X X X
Selection of the Painlevé-I solution
The double-scaling analysis above determines the differential equation
y ′ ′ = 6 y 2 + t , y'' = 6y^2 + t, y ′′ = 6 y 2 + t , but not its solution. This distinction is essential here. Different solutions can have the same algebraic asymptotic expansion while differing by exponentially small terms, and on the E < E ∗ E < E^* E < E ∗ Γ − \Gamma_- Γ − N N N
To make this identification we use the complex cubic matrix model studied by Bleher and Deaño [57 , 58 ] . Their potential can be written
V B D ( z ; u ) = z 2 2 − u z 3 . (221) V_{\mathrm{BD}}(z;u) = \frac{z^2}{2} - uz^3. \tag*{(221)} V BD ( z ; u ) = 2 z 2 − u z 3 . (221) For E < 0 E < 0 E < 0 V E ( λ ) V_E(\lambda) V E ( λ )
λ = B E + A E z , B E = − E , A E = − J 2 ( − E ) 1 / 4 . (222) \lambda= B_E + A_Ez,\qquad B_E = \sqrt{-E},\qquad A_E = -\frac{J}{2(-E)^{1/4}}. \tag*{(222)} λ = B E + A E z , B E = − E , A E = − 2 ( − E ) 1/4 J . (222) Indeed, using B E 2 + E = 0 B_E^2 + E = 0 B E 2 + E = 0 A E 2 = J 2 / ( 4 B E ) A_E^2 = J^2/(4B_E) A E 2 = J 2 / ( 4 B E )
V E ( B E + A E z ) = const + z 2 2 − u ( E ) z 3 , u ( E ) = J 12 ( − E ) 3 / 4 . (223) V_E(B_E + A_Ez) = \operatorname{const} + \frac{z^2}{2} - u(E)z^3,\qquad u(E) = \frac{J}{12(-E)^{3/4}}. \tag*{(223)} V E ( B E + A E z ) = const + 2 z 2 − u ( E ) z 3 , u ( E ) = 12 ( − E ) 3/4 J . (223) Thus our outgoing cubic integral is, apart from an E E E [57 , 58 ] .
Bleher and Deaño found a critical coupling
u c = 3 1 / 4 18 . (224) u_c = \frac{3^{1/4}}{18}. \tag*{(224)} u c = 18 3 1/4 . (224) At our edge E ∗ = − 3 s J 2 / 4 E^* = -3s_J^2/4 E ∗ = − 3 s J 2 /4 s J = ( 2 J 2 ) 1 / 3 s_J = (2J^2)^{1/3} s J = ( 2 J 2 ) 1/3 (223) gives precisely
u ( E ∗ ) = u c . (225) u(E^*) = u_c. \tag*{(225)} u ( E ∗ ) = u c . (225) Hence the critical point found from our planar string equations is exactly the critical point of their cubic model.
The agreement also fixes the normalization of the double-scaling coordinate. In the notation of Ref. [57 ] , the Painlevé variable λ B D \lambda_{\mathrm{BD}} λ BD
N 4 / 5 ( u − u c ) = c 1 λ B D , c 1 = 2 − 12 / 5 3 − 7 / 4 . (226) N^{4/5}(u-u_c)=c_1\lambda_{\mathrm{BD}},\qquad c_1=2^{-12/5}3^{-7/4}. \tag*{(226)} N 4/5 ( u − u c ) = c 1 λ BD , c 1 = 2 − 12/5 3 − 7/4 . (226) Expanding Eq. (223) for E = E ∗ + s J 2 N − 4 / 5 X E=E^*+s_J^2N^{-4/5}X E = E ∗ + s J 2 N − 4/5 X
N 4 / 5 ( u − u c ) = u c X + o ( 1 ) . (227) N^{4/5}(u-u_c)=u_cX+o(1). \tag*{(227)} N 4/5 ( u − u c ) = u c X + o ( 1 ) . (227) Since u c / c 1 = 2 7 / 5 u_c/c_1=2^{7/5} u c / c 1 = 2 7/5
λ B D = 2 7 / 5 X . (228) \lambda_{\mathrm{BD}}=2^{7/5}X. \tag*{(228)} λ BD = 2 7/5 X . (228) This is exactly the value t = 2 7 / 5 X t=2^{7/5}X t = 2 7/5 X n = N n=N n = N
It remains to identify the contour. Bleher and Deaño define the cubic model on admissible complex contours connecting decay sectors of the cubic exponential. Their family may be parametrized by a parameter α \alpha α α = 1 \alpha=1 α = 1 Γ 0 \Gamma_0 Γ 0 y 1 y_1 y 1 [57 ] . Under Eq. (222) , since A E < 0 A_E<0 A E < 0
Γ − : e − 2 π i / 3 ∞ ⟶ + ∞ (229) \Gamma_-:\ e^{-2\pi i/3}\infty\longrightarrow+\infty \tag*{(229)} Γ − : e − 2 π i /3 ∞ ⟶ + ∞ (229) is mapped to
e i π / 3 ∞ ⟶ − ∞ . (230) e^{i\pi/3}\infty\longrightarrow-\infty. \tag*{(230)} e iπ /3 ∞ ⟶ − ∞. (230) The ray arg z = π / 3 \arg z=\pi/3 arg z = π /3 Γ 0 \Gamma_0 Γ 0 − 1 -1 − 1 N N N E E E ( − 1 ) N (-1)^N ( − 1 ) N α = 1 \alpha=1 α = 1
Consequently the Painlevé solution entering the physical E + i 0 E+i0 E + i 0
y ( t ) = y 1 ( t ) , (231) y(t)=y_1(t), \tag*{(231)} y ( t ) = y 1 ( t ) , (231) the tritronquée solution selected by that contour. The conjugate contour Γ − ‾ \overline{\Gamma_-} Γ − E − i 0 E-i0 E − i 0 N N N
The important point is that Eq. (211) determines only the local critical equation, whereas the original Riccati boundary condition fixes the global contour and therefore the particular solution of that equation. The asymptotic consequences of this choice are discussed in the main text.
Painlevé-I tau-function interpretation
The same result has a useful partition-function formulation. Let F ( X ) \mathcal{F}(X) F ( X ) O ( 1 ) O(1) O ( 1 ) log Z N ( − ) \log\mathcal{Z}_N^{(-)} log Z N ( − ) N N N (179) , the endpoint scaling Eq. (216) , and the physical selection y = y 1 y=y_1 y = y 1 (231) , one may define the associated Painlevé-I tau function by
d 2 d t 2 log τ P I ( 1 ) ( t ) = − y 1 ( t ) , F ( X ) = log τ P I ( 1 ) ( 2 7 / 5 X ) . (232) \frac{\mathrm{d}^2}{\mathrm{d}t^2}\log\tau_{\mathrm{PI}}^{(1)}(t)=-y_1(t),\qquad\mathcal{F}(X)=\log\tau_{\mathrm{PI}}^{(1)}(2^{7/5}X). \tag*{(232)} d t 2 d 2 log τ PI ( 1 ) ( t ) = − y 1 ( t ) , F ( X ) = log τ PI ( 1 ) ( 2 7/5 X ) . (232) The second relation is understood up to an affine function of t t t Ξ N = − T E log Z N ( − ) \Xi_N=-\mathcal{T}_E\log\mathcal{Z}_N^{(-)} Ξ N = − T E log Z N ( − ) ν N = − π − 1 Im Ξ N \nu_N=-\pi^{-1}\operatorname{Im}\Xi_N ν N = − π − 1 Im Ξ N
Φ ( X ) = 2 − 3 / 5 π Im d d t log τ P I ( 1 ) ( t ) , t = 2 7 / 5 X . (233) \Phi(X)=\frac{2^{-3/5}}{\pi}\operatorname{Im}\frac{\mathrm{d}}{\mathrm{d}t}\log\tau_{\mathrm{PI}}^{(1)}(t),\qquad t=2^{7/5}X. \tag*{(233)} Φ ( X ) = π 2 − 3/5 Im d t d log τ PI ( 1 ) ( t ) , t = 2 7/5 X . (233) Differentiating Eq. (233) reproduces Eq. (218) ; the tau-function formulation therefore encodes the same critical result directly at the level of the partition function.
Thouless heuristics for critical scaling identifications
The exponents of the critical regime, E − E ∗ ∼ N − 4 / 5 E-E^* \sim N^{-4/5} E − E ∗ ∼ N − 4/5 ν N ∼ N − 1 / 5 \nu_N \sim N^{-1/5} ν N ∼ N − 1/5 ρ N ∼ N 3 / 5 \rho_N \sim N^{3/5} ρ N ∼ N 3/5 5 / 4 5/4 5/4 N − 2 / 3 N^{-2/3} N − 2/3 N N N N = ∞ N=\infty N = ∞ N N N s J 2 s_J^2 s J 2 s J − 1 s_J^{-1} s J − 1 δ = E − E ∗ \delta=E-E^* δ = E − E ∗
Why the Gaussian counting argument has to be modified
For an N × N N \times N N × N ρ ( λ ) ≃ π − 1 2 − λ \rho(\lambda)\simeq\pi^{-1}\sqrt{2-\lambda} ρ ( λ ) ≃ π − 1 2 − λ δ \delta δ N ∫ 2 − δ 2 ρ d λ ∼ N δ 3 / 2 ∼ 1 N\int_{2-\delta}^{2}\rho\,d\lambda\sim N\delta^{3/2}\sim1 N ∫ 2 − δ 2 ρ d λ ∼ N δ 3/2 ∼ 1 δ ∼ N − 2 / 3 \delta\sim N^{-2/3} δ ∼ N − 2/3 N N N δ \delta δ W − 4 / 5 W^{-4/5} W − 4/5 M − 2 / 3 M^{-2/3} M − 2/3
Mean-field Green function and the planar cubic
At N = ∞ N=\infty N = ∞ (1) is obtained in the self-consistent Born approximation. Since the variance of each matrix element of W ( x ) W(x) W ( x ) 2 T = J 2 / N 2T=J^2/N 2 T = J 2 / N 1 / N 1/N 1/ N δ \delta δ x x x g = N − 1 Tr G ‾ ( x , x ) g=N^{-1}\operatorname{Tr}\overline{G}(x,x) g = N − 1 Tr G ( x , x )
Σ ( E ) = 2 T N g = J 2 g , G ‾ ( k ) = 1 E − Σ ( E ) − k 2 . (234) \Sigma(E)=2TN g=J^2g,\qquad\overline{G}(k)=\frac{1}{E-\Sigma(E)-k^2}. \tag*{(234)} Σ ( E ) = 2 T N g = J 2 g , G ( k ) = E − Σ ( E ) − k 2 1 . (234) Note that the factor 1 / N 1/N 1/ N N N N Σ \Sigma Σ N N N E E E E − Σ E-\Sigma E − Σ E − Σ = − κ 2 E-\Sigma=-\kappa^2 E − Σ = − κ 2 Re κ > 0 \operatorname{Re}\kappa>0 Re κ > 0 G ‾ ( x , x ′ ) = − e − κ ∣ x − x ′ ∣ / ( 2 κ ) \overline{G}(x,x')=-e^{-\kappa|x-x'|}/(2\kappa) G ( x , x ′ ) = − e − κ ∣ x − x ′ ∣ / ( 2 κ ) g = − 1 / ( 2 κ ) g=-1/(2\kappa) g = − 1/ ( 2 κ ) − κ 2 = E + J 2 / ( 2 κ ) -\kappa^2=E+J^2/(2\kappa) − κ 2 = E + J 2 / ( 2 κ )
κ 3 + E κ + J 2 2 = 0 , (235) \kappa^3+E\kappa+\frac{J^2}{2}=0, \tag*{(235)} κ 3 + E κ + 2 J 2 = 0 , (235) which is the planar cubic (S148) with d ≡ κ d\equiv\kappa d ≡ κ r = J 2 / ( 4 d ) r=J^2/(4d) r = J 2 / ( 4 d ) ρ N = ( π T ) − 1 Im r N , N \rho_N=(\pi T)^{-1}\operatorname{Im}r_{N,N} ρ N = ( π T ) − 1 Im r N , N ρ N = N ρ 1 \rho_N=N\rho_1 ρ N = N ρ 1 ρ 1 = − π − 1 Im g = ∣ Im κ ∣ / ( 2 π ∣ κ ∣ 2 ) \rho_1=-\pi^{-1}\operatorname{Im}g=|\operatorname{Im}\kappa|/(2\pi|\kappa|^2) ρ 1 = − π − 1 Im g = ∣ Im κ ∣/ ( 2 π ∣ κ ∣ 2 ) d d d
The edge is the double root of (S199), κ ∗ = s J / 2 \kappa_* = s_J/2 κ ∗ = s J /2 E ∗ = − 3 κ ∗ 2 E^*=-3\kappa_*^2 E ∗ = − 3 κ ∗ 2 E − Re Σ = − κ ∗ 2 < 0 E-\operatorname{Re}\Sigma=-\kappa_*^2<0 E − Re Σ = − κ ∗ 2 < 0 κ ∗ − 1 \kappa_*^{-1} κ ∗ − 1 δ > 0 \delta>0 δ > 0 κ − κ ∗ ≃ − i δ / 3 \kappa-\kappa_*\simeq-\mathrm{i}\sqrt{\delta/3} κ − κ ∗ ≃ − i δ /3
ρ 1 ( E ) ≃ 1 2 π κ ∗ 2 δ 3 , ρ N = N ρ 1 . (236) \rho_1(E)\simeq\frac{1}{2\pi\kappa_*^2}\sqrt{\frac{\delta}{3}},\qquad\rho_N=N\rho_1. \tag*{(236)} ρ 1 ( E ) ≃ 2 π κ ∗ 2 1 3 δ , ρ N = N ρ 1 . (236) In the scaling variable δ = s J 2 N − 4 / 5 X \delta=s_J^2N^{-4/5}X δ = s J 2 N − 4/5 X X X X (31) of the critical density (30) .
Scattering rate and diffusion along the wire
The imaginary part of the self-energy is the golden-rule rate of scattering out of a given channel,
1 τ = − 2 Im Σ = 2 π J 2 ρ 1 . (237) \frac{1}{\tau}=-2\operatorname{Im}\Sigma=2\pi J^{2}\rho_{1}. \tag*{(237)} τ 1 = − 2 Im Σ = 2 π J 2 ρ 1 . (237) A particle scatters into each of the N N N J 2 / N J^{2}/N J 2 / N ρ 1 \rho_{1} ρ 1 N N N δ \sqrt{\delta} δ
The diffusion constant along the wire follows from the Kubo formula. The current vertex 2 k 2k 2 k k k k G ‾ \overline{G} G N = ∞ N=\infty N = ∞ G ‾ ( k ) ≃ 2 κ ∗ Im κ / ( κ ∗ 2 + k 2 ) \overline{G}(k)\simeq2\kappa_{*}\operatorname{Im}\kappa/(\kappa_{*}^{2}+k^{2}) G ( k ) ≃ 2 κ ∗ Im κ / ( κ ∗ 2 + k 2 ) σ 1 = π − 1 ∫ d k 2 π ( 2 k ) 2 [ Im G ‾ ( k ) ] 2 = ( Im κ ) 2 / ( 2 π κ ∗ 3 ) \sigma_{1}=\pi^{-1}\int\frac{dk}{2\pi}(2k)^{2}[\operatorname{Im}\overline{G}(k)]^{2}=(\operatorname{Im}\kappa)^{2}/(2\pi\kappa_{*}^{3}) σ 1 = π − 1 ∫ 2 π d k ( 2 k ) 2 [ Im G ( k ) ] 2 = ( Im κ ) 2 / ( 2 π κ ∗ 3 ) σ 1 = ρ 1 D \sigma_{1}=\rho_{1}D σ 1 = ρ 1 D
D = ∣ Im κ ∣ κ ∗ = 2 π κ ∗ ρ 1 ≃ 1 κ ∗ δ 3 . (238) D=\frac{|\operatorname{Im}\kappa|}{\kappa_{*}}=2\pi\kappa_{*}\rho_{1}\simeq\frac{1}{\kappa_{*}}\sqrt{\frac{\delta}{3}}. \tag*{(238)} D = κ ∗ ∣ Im κ ∣ = 2 π κ ∗ ρ 1 ≃ κ ∗ 1 3 δ . (238) Two features are essential. First, D D D N N N D D D δ \sqrt{\delta} δ τ ∼ δ − 1 / 2 \tau\sim\delta^{-1/2} τ ∼ δ − 1/2 D τ ∼ κ ∗ − 1 \sqrt{D\tau}\sim\kappa_{*}^{-1} D τ ∼ κ ∗ − 1
Localization length
In a quasi-one-dimensional wire the localization length is the length on which the dimensionless Thouless conductance g ( L ) = E T h ( L ) / Δ ( L ) g(L)=E_{\mathrm{Th}}(L)/\Delta(L) g ( L ) = E Th ( L ) /Δ ( L ) [4 , 17 , 25 ] . Here E T h ( L ) = D / L 2 E_{\mathrm{Th}}(L)=D/L^{2} E Th ( L ) = D / L 2 L L L Δ ( L ) = 1 / ( ρ N L ) \Delta(L)=1/(\rho_{N}L) Δ ( L ) = 1/ ( ρ N L ) ρ N = N ρ 1 \rho_{N}=N\rho_{1} ρ N = N ρ 1 g ( L ) = ρ N D / L g(L)=\rho_{N}D/L g ( L ) = ρ N D / L
L l o c ( E ) ∼ ρ N D = N ρ 1 D ∼ N δ . (239) L_{\mathrm{loc}}(E)\sim\rho_{N}D=N\rho_{1}D\sim N\delta. \tag*{(239)} L loc ( E ) ∼ ρ N D = N ρ 1 D ∼ N δ . (239) Deep in the band the same formula gives the familiar L l o c ∼ N ℓ L_{\mathrm{loc}}\sim N\ell L loc ∼ N ℓ ℓ \ell ℓ δ \delta δ δ \sqrt{\delta} δ
The critical window
To resolve the structure of the spectrum on the energy scale δ \delta δ t δ ∼ 1 / δ t_{\delta}\sim1/\delta t δ ∼ 1/ δ
L δ = D t δ ∼ δ − 1 / 4 , (240) L_{\delta}=\sqrt{Dt_{\delta}}\sim\delta^{-1/4}, \tag*{(240)} L δ = D t δ ∼ δ − 1/4 , (240) again independent of N N N δ ≪ 1 \delta\ll1 δ ≪ 1 N N N L δ L_{\delta} L δ δ \delta δ
n ( δ ) = L δ N ∫ E ∗ E ∗ + δ ρ 1 ( E ) d E ∼ δ − 1 / 4 N δ 3 / 2 = N δ 5 / 4 , (241) n(\delta)=L_{\delta}N\int_{E_{*}}^{E_{*}+\delta}\rho_{1}(E)\,dE\sim\delta^{-1/4}N\delta^{3/2}=N\delta^{5/4}, \tag*{(241)} n ( δ ) = L δ N ∫ E ∗ E ∗ + δ ρ 1 ( E ) d E ∼ δ − 1/4 N δ 3/2 = N δ 5/4 , (241) in place of N δ 3 / 2 N\delta^{3/2} N δ 3/2 δ − 1 / 4 \delta^{-1/4} δ − 1/4 δ \delta δ
n ( δ ) ∼ 1 ⟺ E − E ∗ ∼ s J 2 N − 4 / 5 , (242) n(\delta)\sim1\quad\Longleftrightarrow\quad E-E^{*}\sim s_{J}^{2}N^{-4/5}, \tag*{(242)} n ( δ ) ∼ 1 ⟺ E − E ∗ ∼ s J 2 N − 4/5 , (242) in agreement with the double-scaling analysis. In terms of the scaling variable, n ( δ ) ∼ ∣ X ∣ 5 / 4 n(\delta)\sim|X|^{5/4} n ( δ ) ∼ ∣ X ∣ 5/4
The same condition has a second, equivalent reading. Setting L δ ∼ L l o c ( δ ) L_{\delta}\sim L_{\mathrm{loc}}(\delta) L δ ∼ L loc ( δ ) δ − 1 / 4 ∼ N δ \delta^{-1/4}\sim N\delta δ − 1/4 ∼ N δ N δ 5 / 4 ∼ 1 N\delta^{5/4}\sim1 N δ 5/4 ∼ 1 E T h ( L δ ) = δ E_{\mathrm{Th}}(L_{\delta})=\delta E Th ( L δ ) = δ Δ ( L δ ) = δ \Delta(L_\delta)=\delta Δ ( L δ ) = δ δ ≫ N − 4 / 5 \delta\gg N^{-4/5} δ ≫ N − 4/5 N δ 3 / 2 ∼ 1 N\delta^{3/2}\sim1 N δ 3/2 ∼ 1
The remaining critical exponents follow at once. At the window the integrated density of states is ν N ∼ N δ 3 / 2 ∼ N − 1 / 5 \nu_N\sim N\delta^{3/2}\sim N^{-1/5} ν N ∼ N δ 3/2 ∼ N − 1/5 ρ N ∼ N δ 1 / 2 ∼ N 3 / 5 \rho_N\sim N\delta^{1/2}\sim N^{3/5} ρ N ∼ N δ 1/2 ∼ N 3/5 (30) . The edge states extend over L δ ∼ L l o c ∼ N 1 / 5 L_\delta\sim L_{\mathrm{loc}}\sim N^{1/5} L δ ∼ L loc ∼ N 1/5 N ⋅ N 1 / 5 = N 6 / 5 N\cdot N^{1/5}=N^{6/5} N ⋅ N 1/5 = N 6/5 d − d ∗ = κ − κ ∗ ∼ δ ∼ N − 2 / 5 d-d_*=\kappa-\kappa_*\sim\sqrt{\delta}\sim N^{-2/5} d − d ∗ = κ − κ ∗ ∼ δ ∼ N − 2/5 A ∼ N − 2 / 5 A\sim N^{-2/5} A ∼ N − 2/5 N 1 / 5 N^{1/5} N 1/5 L δ L_\delta L δ
On the forbidden side the quantity n ( ∣ δ ∣ ) n(|\delta|) n ( ∣ δ ∣ ) exp [ − 4 3 N δ 3 / 2 ] \exp[-\frac{4}{3}N\delta^{3/2}] exp [ − 3 4 N δ 3/2 ] δ \delta δ − log ρ N ∼ N ∣ δ ∣ 5 / 4 -\log\rho_N\sim N|\delta|^{5/4} − log ρ N ∼ N ∣ δ ∣ 5/4 (32) and of the single-eigenvalue activation barrier of Ref. [30 ] . We stress that this last step is an observation about the structure of the result rather than an independent derivation.
The status of the ingredients is as follows. The planar Green function, the cubic (S199) and the edge density (S200) are exact at N = ∞ N=\infty N = ∞ N = ∞ N=\infty N = ∞ 1 / N 1/N 1/ N N N N
Random banded matrices
The same reasoning applies to M × M M\times M M × M W ≫ 1 W\gg1 W ≫ 1 2 2 2 ∼ 1 / W \sim1/W ∼ 1/ W ρ ∝ δ \rho\propto\sqrt{\delta} ρ ∝ δ δ = 2 − λ \delta=2-\lambda δ = 2 − λ ∼ W \sim W ∼ W ∼ 1 / W \sim1/W ∼ 1/ W ρ \rho ρ ∼ W \sim W ∼ W D ∼ W 2 ρ ∝ W 2 δ D\sim W^2\rho\propto W^2\sqrt{\delta} D ∼ W 2 ρ ∝ W 2 δ L l o c ∼ ρ D ∼ W 2 ρ 2 L_{\mathrm{loc}}\sim\rho D\sim W^2\rho^2 L loc ∼ ρ D ∼ W 2 ρ 2 L l o c ∝ W 2 L_{\mathrm{loc}}\propto W^2 L loc ∝ W 2 [61 ] . A state resolving the window δ \delta δ
L δ = D / δ ∼ W δ − 1 / 4 (243) L_\delta=\sqrt{D/\delta}\sim W\delta^{-1/4} \tag*{(243)} L δ = D / δ ∼ W δ − 1/4 (243) sites, and the number of levels within δ \delta δ L δ δ 3 / 2 ∼ W δ 5 / 4 L_\delta\delta^{3/2}\sim W\delta^{5/4} L δ δ 3/2 ∼ W δ 5/4 δ ∼ W − 4 / 5 \delta\sim W^{-4/5} δ ∼ W − 4/5 [65 , 66 ] , and, as before, the equivalent condition L δ ∼ L l o c ( δ ) L_\delta\sim L_{\mathrm{loc}}(\delta) L δ ∼ L loc ( δ ) L δ ∼ W ⋅ W 1 / 5 = W 6 / 5 L_\delta\sim W\cdot W^{1/5}=W^{6/5} L δ ∼ W ⋅ W 1/5 = W 6/5 W 6 / 5 M − 1 W^{6/5}M^{-1} W 6/5 M − 1 σ 2 ( ξ ) \sigma_2(\xi) σ 2 ( ξ ) σ 2 \sigma_2 σ 2 M ≪ W 6 / 5 M\ll W^{6/5} M ≪ W 6/5 M M M M δ 3 / 2 ∼ 1 M\delta^{3/2}\sim1 M δ 3/2 ∼ 1 M − 2 / 3 M^{-2/3} M − 2/3 W W W N N N W W W N N N
In this section we obtain series for the IDOS and the total Lyapunov exponent for large ∣ E ∣ |E| ∣ E ∣ N N N E > 0 E>0 E > 0 E < 0 E<0 E < 0 N = 1 , 2 N=1,2 N = 1 , 2 (23) in the main text.
Explicit results for N = 1 N=1 N = 1 N = 2 N=2 N = 2
Set s = − E / T 2 / 3 s=-E/T^{2/3} s = − E / T 2/3 A = Ai ( s ) A=\operatorname{Ai}(s) A = Ai ( s ) B = Bi ( s ) B=\operatorname{Bi}(s) B = Bi ( s ) s s s T = J 2 / ( 2 N ) T=J^2/(2N) T = J 2 / ( 2 N )
For N = 1 N=1 N = 1 (22) gives τ 1 = A − i B \tau_1=A-iB τ 1 = A − i B A B ′ − A ′ B = 1 / π AB'-A'B=1/\pi A B ′ − A ′ B = 1/ π (23) yields
Γ 1 ( E ) = T 1 / 3 A A ′ + B B ′ A 2 + B 2 , ν 1 ( E ) = T 1 / 3 π 2 ( A 2 + B 2 ) . (244) \Gamma_1(E)=T^{1/3}\frac{AA'+BB'}{A^2+B^2},\qquad\nu_1(E)=\frac{T^{1/3}}{\pi^2(A^2+B^2)}. \tag*{(244)} Γ 1 ( E ) = T 1/3 A 2 + B 2 A A ′ + B B ′ , ν 1 ( E ) = π 2 ( A 2 + B 2 ) T 1/3 . (244) These are Halperin’s exact scalar IDOS and Lyapunov exponent [2 , 6 ] .
For N = 2 N=2 N = 2 τ 2 = s ( A − i B ) 2 − ( A ′ − i B ′ ) 2 \tau_2=s(A-iB)^2-(A'-iB')^2 τ 2 = s ( A − i B ) 2 − ( A ′ − i B ′ ) 2 τ 2 ′ = ( A − i B ) 2 \tau'_2=(A-iB)^2 τ 2 ′ = ( A − i B ) 2 Ξ 2 = T 1 / 3 ( A − i B ) 2 / τ 2 \Xi_2=T^{1/3}(A-iB)^2/\tau_2 Ξ 2 = T 1/3 ( A − i B ) 2 / τ 2
D 2 ( s ) : = ∣ τ 2 ∣ 2 = [ s ( A 2 − B 2 ) − ( A ′ 2 − B ′ 2 ) ] 2 + 4 ( s A B − A ′ B ′ ) 2 . (245) D_2(s):=|\tau_2|^2=\left[s(A^2-B^2)-(A'^2-B'^2)\right]^2+4(sAB-A'B')^2. \tag*{(245)} D 2 ( s ) := ∣ τ 2 ∣ 2 = [ s ( A 2 − B 2 ) − ( A ′2 − B ′2 ) ] 2 + 4 ( s A B − A ′ B ′ ) 2 . (245) Taking real and imaginary parts gives
Γ 2 ( E ) = T 1 / 3 s ( A 2 + B 2 ) 2 − ( A A ′ + B B ′ ) 2 + π − 2 D 2 ( s ) , ν 2 ( E ) = 2 T 1 / 3 π 2 A A ′ + B B ′ D 2 ( s ) . (246) \Gamma_2(E)=T^{1/3}\frac{s(A^2+B^2)^2-(AA'+BB')^2+\pi^{-2}}{D_2(s)},\qquad\nu_2(E)=\frac{2T^{1/3}}{\pi^2}\frac{AA'+BB'}{D_2(s)}. \tag*{(246)} Γ 2 ( E ) = T 1/3 D 2 ( s ) s ( A 2 + B 2 ) 2 − ( A A ′ + B B ′ ) 2 + π − 2 , ν 2 ( E ) = π 2 2 T 1/3 D 2 ( s ) A A ′ + B B ′ . (246) As an independent algebraic check, multiplying Ξ 2 / T 1 / 3 \Xi_2/T^{1/3} Ξ 2 / T 1/3 τ ‾ 2 / τ ‾ 2 \overline{\tau}_2/\overline{\tau}_2 τ 2 / τ 2
( A − i B ) 2 τ ‾ 2 = s ( A 2 + B 2 ) 2 − ( A A ′ + B B ′ ) 2 + π − 2 − 2 i π ( A A ′ + B B ′ ) , (247) (A-iB)^2\overline{\tau}_2=s(A^2+B^2)^2-(AA'+BB')^2+\pi^{-2}-\frac{2i}{\pi}(AA'+BB'), \tag*{(247)} ( A − i B ) 2 τ 2 = s ( A 2 + B 2 ) 2 − ( A A ′ + B B ′ ) 2 + π − 2 − π 2 i ( A A ′ + B B ′ ) , (247) whose real and imaginary parts reproduce Eq. (246) . Thus already at N = 2 N=2 N = 2
Perturbative expansion for E > 0 E>0 E > 0
The cubic matrix integral provides a particularly efficient starting point for the high-energy, or weak-disorder, expansion. We use
Ξ N ( E ) = − T ∂ E log Z N ( − ) ( E ) , (248) \Xi_N(E)=-T\partial_E\log Z_N^{(-)}(E), \tag*{(248)} Ξ N ( E ) = − T ∂ E log Z N ( − ) ( E ) , (248) with
Z N ( − ) ( E ) = 1 N ! ∫ Γ − Δ ( λ ) 2 ∏ j = 1 N exp [ − 1 T ( λ j 3 3 + E λ j ) ] d N λ . (249) Z_N^{(-)}(E)=\frac{1}{N!}\int_{\Gamma_-}\Delta(\lambda)^2\prod_{j=1}^{N}\exp\left[-\frac{1}{T}\left(\frac{\lambda_j^3}{3}+E\lambda_j\right)\right]\,\mathrm{d}^N\lambda. \tag*{(249)} Z N ( − ) ( E ) = N ! 1 ∫ Γ − Δ ( λ ) 2 j = 1 ∏ N exp [ − T 1 ( 3 λ j 3 + E λ j ) ] d N λ . (249) For E > 0 E>0 E > 0 k = E k=\sqrt{E} k = E ε = T / E 3 / 2 = T / k 3 ≪ 1 \varepsilon=T/E^{3/2}=T/k^3\ll1 ε = T / E 3/2 = T / k 3 ≪ 1 Γ − \Gamma_- Γ − λ ∗ = − i k \lambda_*=-ik λ ∗ = − ik
λ j = − i k + e i π / 4 T 2 k x j . (250) \lambda_j=-ik+e^{i\pi/4}\sqrt{\frac{T}{2k}}\,x_j. \tag*{(250)} λ j = − ik + e iπ /4 2 k T x j . (250) Since the potential is exactly cubic, the expansion terminates,
− 1 T ( λ j 3 3 + E λ j ) = 2 i k 3 3 T − x j 2 2 + g x j 3 , g = − e 3 i π / 4 3 2 3 / 2 ε , g 2 = − i 72 ε . (251) -\frac{1}{T}\left(\frac{\lambda_j^3}{3}+E\lambda_j\right)=\frac{2ik^3}{3T}-\frac{x_j^2}{2}+gx_j^3,\qquad g=-\frac{e^{3i\pi/4}}{3\,2^{3/2}}\sqrt{\varepsilon},\qquad g^2=-\frac{i}{72}\varepsilon. \tag*{(251)} − T 1 ( 3 λ j 3 + E λ j ) = 3 T 2 i k 3 − 2 x j 2 + g x j 3 , g = − 3 2 3/2 e 3 iπ /4 ε , g 2 = − 72 i ε . (251) Including the Jacobian and the Vandermonde scaling gives
Z N ( − ) = C N e 2 i N k 3 / ( 3 T ) ( T 2 k ) N 2 / 2 ⟨ e g A ⟩ 0 , A : = Tr X 3 , (252) Z_N^{(-)}=C_Ne^{2iNk^3/(3T)}\left(\frac{T}{2k}\right)^{N^2/2}\left\langle e^{gA}\right\rangle_0,\qquad A:=\operatorname{Tr}X^3, \tag*{(252)} Z N ( − ) = C N e 2 i N k 3 / ( 3 T ) ( 2 k T ) N 2 /2 ⟨ e g A ⟩ 0 , A := Tr X 3 , (252) where C N C_N C N E E E ⟨ ⋯ ⟩ 0 \langle\cdots\rangle_0 ⟨ ⋯ ⟩ 0 e − Tr X 2 / 2 e^{-\operatorname{Tr}X^2/2} e − Tr X 2 /2 A A A X → − X X\to-X X → − X
log ⟨ e g A ⟩ 0 = ∑ m ≥ 1 g 2 m ( 2 m ) ! κ 2 m ( A ) . (253) \log\left\langle e^{gA}\right\rangle_0=\sum_{m\geq1}\frac{g^{2m}}{(2m)!}\kappa_{2m}(A). \tag*{(253)} log ⟨ e g A ⟩ 0 = m ≥ 1 ∑ ( 2 m )! g 2 m κ 2 m ( A ) . (253) Because g 2 ∝ − i ε g^2 \propto-i\varepsilon g 2 ∝ − i ε ε \varepsilon ε log Z N ( − ) \log Z_N^{(-)} log Z N ( − ) Γ N \Gamma_N Γ N T T T T 3 T^3 T 3 T 5 T^5 T 5 ν N \nu_N ν N T 0 T^0 T 0 T 2 T^2 T 2 T 4 T^4 T 4
The weak-disorder expansion has a topological organization [76 ] . A connected Wick contraction in κ 2 m \kappa_{2m} κ 2 m 2 m 2m 2 m 3 m 3m 3 m h h h F = m + 2 − 2 h F=m+2-2h F = m + 2 − 2 h N F N^F N F κ 2 m \kappa_{2m} κ 2 m N m + 2 N^{m+2} N m + 2 N m N^m N m g 2 m ∝ N − m g^{2m}\propto N^{-m} g 2 m ∝ N − m E E E J J J g 2 m κ 2 m / ( 2 m ) ! g^{2m}\kappa_{2m}/(2m)! g 2 m κ 2 m / ( 2 m )! N 2 − 2 h N^{2-2h} N 2 − 2 h E E E J J J
κ 2 ( A ) = 3 N ( 4 N 2 + 1 ) , κ 4 ( A ) = 648 N 2 ( 8 N 2 + 7 ) , κ 6 ( A ) = 29160 N ( 336 N 4 + 664 N 2 + 105 ) . (254) \begin{aligned}
\kappa_2(A) &= 3N(4N^2+1),\\
\kappa_4(A) &= 648N^2(8N^2+7),\\
\kappa_6(A) &= 29160N(336N^4+664N^2+105). \tag*{(254)}
\end{aligned} κ 2 ( A ) κ 4 ( A ) κ 6 ( A ) = 3 N ( 4 N 2 + 1 ) , = 648 N 2 ( 8 N 2 + 7 ) , = 29160 N ( 336 N 4 + 664 N 2 + 105 ) . (254) Substitution into Eq. (253) gives
log ⟨ e g A ⟩ 0 = − i N ( 4 N 2 + 1 ) 48 ε − N 2 ( 8 N 2 + 7 ) 192 ε 2 + i N ( 336 N 4 + 664 N 2 + 105 ) 9216 ε 3 + O ( ε 4 ) . (255) \log\langle e^{gA}\rangle_0=-\frac{iN(4N^2+1)}{48}\varepsilon-\frac{N^2(8N^2+7)}{192}\varepsilon^2+\frac{iN(336N^4+664N^2+105)}{9216}\varepsilon^3+O(\varepsilon^4). \tag*{(255)} log ⟨ e g A ⟩ 0 = − 48 i N ( 4 N 2 + 1 ) ε − 192 N 2 ( 8 N 2 + 7 ) ε 2 + 9216 i N ( 336 N 4 + 664 N 2 + 105 ) ε 3 + O ( ε 4 ) . (255) Hence, up to an E E E
log Z N ( − ) = 2 i N E 3 / 2 3 T + N 2 2 log ( T 2 E ) − i N ( 4 N 2 + 1 ) 48 ε − N 2 ( 8 N 2 + 7 ) 192 ε 2 + i N ( 336 N 4 + 664 N 2 + 105 ) 9216 ε 3 + O ( ε 4 ) . (256) \log Z_N^{(-)}=\frac{2iNE^{3/2}}{3T}+\frac{N^2}{2}\log\left(\frac{T}{2\sqrt{E}}\right)-\frac{iN(4N^2+1)}{48}\varepsilon-\frac{N^2(8N^2+7)}{192}\varepsilon^2+\frac{iN(336N^4+664N^2+105)}{9216}\varepsilon^3+O(\varepsilon^4). \tag*{(256)} log Z N ( − ) = 3 T 2 i N E 3/2 + 2 N 2 log ( 2 E T ) − 48 i N ( 4 N 2 + 1 ) ε − 192 N 2 ( 8 N 2 + 7 ) ε 2 + 9216 i N ( 336 N 4 + 664 N 2 + 105 ) ε 3 + O ( ε 4 ) . (256) Using ∂ E ε m = − 3 m ε m / ( 2 E ) \partial_E\varepsilon^m=-3m\varepsilon^m/(2E) ∂ E ε m = − 3 m ε m / ( 2 E ) (248) , we obtain
Ξ N ( E ) = − i N E + N 2 T 4 E − i N ( 4 N 2 + 1 ) T 2 32 E 5 / 2 − N 2 ( 8 N 2 + 7 ) T 3 64 E 4 + i N ( 336 N 4 + 664 N 2 + 105 ) T 4 2048 E 11 / 2 + O ( T 5 E 7 ) . (257) \Xi_N(E)=-iN\sqrt{E}+\frac{N^2T}{4E}-\frac{iN(4N^2+1)T^2}{32E^{5/2}}-\frac{N^2(8N^2+7)T^3}{64E^4}+\frac{iN(336N^4+664N^2+105)T^4}{2048E^{11/2}}+O\left(\frac{T^5}{E^7}\right). \tag*{(257)} Ξ N ( E ) = − i N E + 4 E N 2 T − 32 E 5/2 i N ( 4 N 2 + 1 ) T 2 − 64 E 4 N 2 ( 8 N 2 + 7 ) T 3 + 2048 E 11/2 i N ( 336 N 4 + 664 N 2 + 105 ) T 4 + O ( E 7 T 5 ) . (257) Since Ξ N = Γ N − i π ν N \Xi_N=\Gamma_N-i\pi\nu_N Ξ N = Γ N − iπ ν N
Γ N ( E ) = N 2 T 4 E [ 1 − 8 N 2 + 7 16 ( T E 3 / 2 ) 2 + O ( T 4 E 6 ) ] . (258) \Gamma_N(E)=\frac{N^2T}{4E}\left[1-\frac{8N^2+7}{16}\left(\frac{T}{E^{3/2}}\right)^2+O\left(\frac{T^4}{E^6}\right)\right]. \tag*{(258)} Γ N ( E ) = 4 E N 2 T [ 1 − 16 8 N 2 + 7 ( E 3/2 T ) 2 + O ( E 6 T 4 ) ] . (258) ν N ( E ) = N E π [ 1 + 4 N 2 + 1 32 ( T E 3 / 2 ) 2 − 336 N 4 + 664 N 2 + 105 2048 ( T E 3 / 2 ) 4 + O ( T 6 E 9 ) ] . (259) \nu_N(E)=\frac{N\sqrt{E}}{\pi}\left[1+\frac{4N^2+1}{32}\left(\frac{T}{E^{3/2}}\right)^2-\frac{336N^4+664N^2+105}{2048}\left(\frac{T}{E^{3/2}}\right)^4+O\left(\frac{T^6}{E^9}\right)\right]. \tag*{(259)} ν N ( E ) = π N E [ 1 + 32 4 N 2 + 1 ( E 3/2 T ) 2 − 2048 336 N 4 + 664 N 2 + 105 ( E 3/2 T ) 4 + O ( E 9 T 6 ) ] . (259) For N = 1 N=1 N = 1
ν 1 ( E ) = E π [ 1 + 5 32 ( T E 3 / 2 ) 2 − 1105 2048 ( T E 3 / 2 ) 4 + ⋯ ] . (260) \nu_1(E)=\frac{\sqrt{E}}{\pi}\left[1+\frac{5}{32}\left(\frac{T}{E^{3/2}}\right)^2-\frac{1105}{2048}\left(\frac{T}{E^{3/2}}\right)^4+\cdots\right]. \tag*{(260)} ν 1 ( E ) = π E [ 1 + 32 5 ( E 3/2 T ) 2 − 2048 1105 ( E 3/2 T ) 4 + ⋯ ] . (260) in agreement with the large-negative-s s s Tr X 3 \operatorname{Tr}X^3 Tr X 3
Negative-energy expansion from the Airy determinant
For E < 0 E<0 E < 0 z = ∣ E ∣ / T 2 / 3 z=|E|/T^{2/3} z = ∣ E ∣/ T 2/3 ε = z − 3 / 2 = T / ∣ E ∣ 3 / 2 ≪ 1 \varepsilon=z^{-3/2}=T/|E|^{3/2}\ll1 ε = z − 3/2 = T /∣ E ∣ 3/2 ≪ 1 S ( z ) = 4 z 3 / 2 / 3 = 4 / ( 3 ε ) = 4 ∣ E ∣ 3 / 2 / ( 3 T ) S(z)=4z^{3/2}/3=4/(3\varepsilon)=4|E|^{3/2}/(3T) S ( z ) = 4 z 3/2 /3 = 4/ ( 3 ε ) = 4∣ E ∣ 3/2 / ( 3 T )
τ N ( z ) = det 0 ≤ j , k ≤ N − 1 [ Ai ( j + k ) ( z ) − i Bi ( j + k ) ( z ) ] . (261) \tau_N(z)=\det_{0\le j,k\le N-1}\left[\operatorname{Ai}^{(j+k)}(z)-i\operatorname{Bi}^{(j+k)}(z)\right]. \tag*{(261)} τ N ( z ) = 0 ≤ j , k ≤ N − 1 det [ Ai ( j + k ) ( z ) − i Bi ( j + k ) ( z ) ] . (261) Introduce the real Hankel matrices
A N ( z ) = [ Ai ( j + k ) ( z ) ] j , k = 0 N − 1 , B N ( z ) = [ Bi ( j + k ) ( z ) ] j , k = 0 N − 1 . (262) A_N(z)=\left[\operatorname{Ai}^{(j+k)}(z)\right]_{j,k=0}^{N-1},\qquad B_N(z)=\left[\operatorname{Bi}^{(j+k)}(z)\right]_{j,k=0}^{N-1}. \tag*{(262)} A N ( z ) = [ Ai ( j + k ) ( z ) ] j , k = 0 N − 1 , B N ( z ) = [ Bi ( j + k ) ( z ) ] j , k = 0 N − 1 . (262) Then
τ N ( z ) = ( − i ) N det B N ( z ) det [ I + i X N ( z ) ] , X N : = B N − 1 A N . (263) \tau_N(z)=(-i)^N\det B_N(z)\det\left[I+iX_N(z)\right],\qquad X_N:=B_N^{-1}A_N. \tag*{(263)} τ N ( z ) = ( − i ) N det B N ( z ) det [ I + i X N ( z ) ] , X N := B N − 1 A N . (263) For z ≫ 1 z\gg1 z ≫ 1
Bi ( z ) ∼ z − 1 / 4 π e 2 z 3 / 2 / 3 , Ai ( z ) ∼ z − 1 / 4 2 π e − 2 z 3 / 2 / 3 , (264) \operatorname{Bi}(z)\sim\frac{z^{-1/4}}{\sqrt{\pi}}e^{2z^{3/2}/3},\qquad\operatorname{Ai}(z)\sim\frac{z^{-1/4}}{2\sqrt{\pi}}e^{-2z^{3/2}/3}, \tag*{(264)} Bi ( z ) ∼ π z − 1/4 e 2 z 3/2 /3 , Ai ( z ) ∼ 2 π z − 1/4 e − 2 z 3/2 /3 , (264) and similarly for their derivatives, so that X N = O ( e − S ) X_N=O(e^{-S}) X N = O ( e − S )
log τ N = − i π N 2 + log det B N + Tr log ( I + i X N ) = − i π N 2 + log det B N + i Tr X N + 1 2 Tr X N 2 − i 3 Tr X N 3 + ⋯ . (265) \begin{aligned}
\log\tau_N&=-\frac{i\pi N}{2}+\log\det B_N+\operatorname{Tr}\log(I+iX_N)\\
&=-\frac{i\pi N}{2}+\log\det B_N+i\operatorname{Tr}X_N+\frac{1}{2}\operatorname{Tr}X_N^2-\frac{i}{3}\operatorname{Tr}X_N^3+\cdots. \tag*{(265)}
\end{aligned} log τ N = − 2 iπ N + log det B N + Tr log ( I + i X N ) = − 2 iπ N + log det B N + i Tr X N + 2 1 Tr X N 2 − 3 i Tr X N 3 + ⋯ . (265) The real all-Bi determinant gives the algebraic expansion of the total Lyapunov exponent. Expanding it to the same order as above gives, recalling that ε = T / ∣ E ∣ 3 / 2 ≪ 1 \varepsilon=T/|E|^{3/2}\ll1 ε = T /∣ E ∣ 3/2 ≪ 1
Γ N ( E ) ∼ N ∣ E ∣ [ 1 − N 4 ε − 4 N 2 + 1 32 ε 2 − N ( 8 N 2 + 7 ) 64 ε 3 − 336 N 4 + 664 N 2 + 105 2048 ε 4 + O ( ε 5 ) ] , (266) \Gamma_N(E)\sim N\sqrt{|E|}\left[1-\frac{N}{4}\varepsilon-\frac{4N^2+1}{32}\varepsilon^2-\frac{N(8N^2+7)}{64}\varepsilon^3-\frac{336N^4+664N^2+105}{2048}\varepsilon^4+O(\varepsilon^5)\right], \tag*{(266)} Γ N ( E ) ∼ N ∣ E ∣ [ 1 − 4 N ε − 32 4 N 2 + 1 ε 2 − 64 N ( 8 N 2 + 7 ) ε 3 − 2048 336 N 4 + 664 N 2 + 105 ε 4 + O ( ε 5 ) ] , (266) up to exponentially small corrections beginning at order e − 2 S e^{-2S} e − 2 S N = 1 N=1 N = 1 z z z T 1 / 3 Bi ′ ( z ) / Bi ( z ) T^{1/3}\operatorname{Bi}'(z)/\operatorname{Bi}(z) T 1/3 Bi ′ ( z ) / Bi ( z ) N = 2 N=2 N = 2 s s s
By contrast, the IDOS is nonperturbative in T T T (265) . At leading exponential order only the one-Ai sector contributes. Defining R N ( z ) : = Tr ( B N − 1 A N ) R_N(z):=\operatorname{Tr}(B_N^{-1}A_N) R N ( z ) := Tr ( B N − 1 A N )
ν N ( E ) = − T 1 / 3 π ∂ z R N ( z ) + O ( e − 3 S ) . (267) \nu_N(E)=-\frac{T^{1/3}}{\pi}\partial_zR_N(z)+O(e^{-3S}). \tag*{(267)} ν N ( E ) = − π T 1/3 ∂ z R N ( z ) + O ( e − 3 S ) . (267) There is no e − 2 S e^{-2S} e − 2 S X N X_N X N (265) are real, whereas odd powers are imaginary.
The one-instanton term can be evaluated systematically from the large-z z z
R N ( z ) = 2 3 N − 4 ( N − 1 ) ! ε − ( N − 1 ) e − 4 / ( 3 ε ) [ 1 + r 1 ( N ) ε + r 2 ( N ) ε 2 + O ( ε 3 ) ] , (268) R_N(z)=\frac{2^{3N-4}}{(N-1)!}\varepsilon^{-(N-1)}e^{-4/(3\varepsilon)}\left[1+r_1(N)\varepsilon+r_2(N)\varepsilon^2+O(\varepsilon^3)\right], \tag*{(268)} R N ( z ) = ( N − 1 )! 2 3 N − 4 ε − ( N − 1 ) e − 4/ ( 3 ε ) [ 1 + r 1 ( N ) ε + r 2 ( N ) ε 2 + O ( ε 3 ) ] , (268) r 1 ( N ) = − 21 N 2 − 51 N + 35 24 , r 2 ( N ) = 441 N 4 − 2880 N 3 + 7059 N 2 − 8340 N + 3745 1152 . (269) r_1(N)=-\frac{21N^2-51N+35}{24},\qquad r_2(N)=\frac{441N^4-2880N^3+7059N^2-8340N+3745}{1152}. \tag*{(269)} r 1 ( N ) = − 24 21 N 2 − 51 N + 35 , r 2 ( N ) = 1152 441 N 4 − 2880 N 3 + 7059 N 2 − 8340 N + 3745 . (269) The leading prefactor has a simple saddle interpretation: the scalar subdominant-to-dominant Airy ratio contributes 1 2 e − S \frac{1}{2}e^{-S} 2 1 e − S ( 8 / ε ) N − 1 (8/\varepsilon)^{N-1} ( 8/ ε ) N − 1 1 / ( N − 1 ) ! 1/(N-1)! 1/ ( N − 1 )!
Differentiating Eq. (268) according to Eq. (267) gives
ν N ( E ) ≃ ∣ E ∣ π 8 N − 1 ( N − 1 ) ! ( ∣ E ∣ 3 / 2 T ) N − 1 exp [ − 4 ∣ E ∣ 3 / 2 3 T ] × [ 1 − 21 N 2 − 33 N + 17 24 T ∣ E ∣ 3 / 2 + 441 N 4 − 2124 N 3 + 3711 N 2 − 3408 N + 1225 1152 ( T ∣ E ∣ 3 / 2 ) 2 + O ( T 3 ∣ E ∣ 9 / 2 ) ] . (270) \begin{aligned}
\nu_N(E)\simeq{}&\frac{\sqrt{|E|}}{\pi}\frac{8^{N-1}}{(N-1)!}\left(\frac{|E|^{3/2}}{T}\right)^{N-1}\exp\left[-\frac{4|E|^{3/2}}{3T}\right]\\
&\times\left[1-\frac{21N^2-33N+17}{24}\frac{T}{|E|^{3/2}}+\frac{441N^4-2124N^3+3711N^2-3408N+1225}{1152}\left(\frac{T}{|E|^{3/2}}\right)^2+O\left(\frac{T^3}{|E|^{9/2}}\right)\right]. \tag*{(270)}
\end{aligned} ν N ( E ) ≃ π ∣ E ∣ ( N − 1 )! 8 N − 1 ( T ∣ E ∣ 3/2 ) N − 1 exp [ − 3 T 4∣ E ∣ 3/2 ] × [ 1 − 24 21 N 2 − 33 N + 17 ∣ E ∣ 3/2 T + 1152 441 N 4 − 2124 N 3 + 3711 N 2 − 3408 N + 1225 ( ∣ E ∣ 3/2 T ) 2 + O ( ∣ E ∣ 9/2 T 3 ) ] . (270) For N = 1 N=1 N = 1
ν 1 ( E ) ≃ ∣ E ∣ π exp [ − 4 ∣ E ∣ 3 / 2 3 T ] [ 1 − 5 24 T ∣ E ∣ 3 / 2 − 155 1152 ( T ∣ E ∣ 3 / 2 ) 2 + ⋯ ] , (271) \nu_1(E) \simeq\frac{\sqrt{|E|}}{\pi}\exp\left[-\frac{4|E|^{3/2}}{3T}\right]\left[1-\frac{5}{24}\frac{T}{|E|^{3/2}}-\frac{155}{1152}\left(\frac{T}{|E|^{3/2}}\right)^2+\cdots\right], \tag*{(271)} ν 1 ( E ) ≃ π ∣ E ∣ exp [ − 3 T 4∣ E ∣ 3/2 ] [ 1 − 24 5 ∣ E ∣ 3/2 T − 1152 155 ( ∣ E ∣ 3/2 T ) 2 + ⋯ ] , (271) in agreement with the direct large-z z z
The determinant representation therefore provides a natural trans-series organization on the E < 0 E<0 E < 0
Asymptotics of the critical scaling functions
We now derive the asymptotic expansions of the critical IDOS scaling function Φ ( X ) \Phi(X) Φ ( X ) (211) , and the outgoing Riccati contour was shown in Eq. (231) to select the tritronquée solution y 1 ( t ) y_1(t) y 1 ( t ) (31) and (32) of the Main Text.
It is useful first to keep the real and imaginary critical observables together. The exact Toda relation (S178) determines the variation of Ξ N \Xi_N Ξ N r N , N r_{N,N} r N , N r ∗ = s J 2 / 4 r_* = s_J^2/4 r ∗ = s J 2 /4 (216) . Upon integration with respect to E E E E = E ∗ E=E^* E = E ∗
Ξ N ( E ) = Ξ N , r e g ( E ) + s J N − 1 / 5 X ( X ) + o ( N − 1 / 5 ) , X ( X ) : = Ψ ( X ) − i π Φ ( X ) , (272) \Xi_N(E)=\Xi_{N,\mathrm{reg}}(E)+s_JN^{-1/5}\mathcal{X}(X)+o(N^{-1/5}),\qquad\mathcal{X}(X):=\Psi(X)-i\pi\Phi(X), \tag*{(272)} Ξ N ( E ) = Ξ N , reg ( E ) + s J N − 1/5 X ( X ) + o ( N − 1/5 ) , X ( X ) := Ψ ( X ) − iπ Φ ( X ) , (272) where Ξ N , r e g ( E ) \Xi_{N,\mathrm{reg}}(E) Ξ N , reg ( E ) (214) , the endpoint expansion (S180), and E = E ∗ + s J 2 N − 4 / 5 X E=E^*+s_J^2N^{-4/5}X E = E ∗ + s J 2 N − 4/5 X
X ′ ( X ) = 2 4 / 5 y 1 ( 2 7 / 5 X ) , Ψ ′ ( X ) = 2 4 / 5 ℜ y 1 ( 2 7 / 5 X ) , Φ ′ ( X ) = − 2 4 / 5 π ℑ y 1 ( 2 7 / 5 X ) . (273) \mathcal{X}'(X)=2^{4/5}y_1(2^{7/5}X),\qquad\Psi'(X)=2^{4/5}\Re y_1(2^{7/5}X),\qquad\Phi'(X)=-\frac{2^{4/5}}{\pi}\Im y_1(2^{7/5}X). \tag*{(273)} X ′ ( X ) = 2 4/5 y 1 ( 2 7/5 X ) , Ψ ′ ( X ) = 2 4/5 ℜ y 1 ( 2 7/5 X ) , Φ ′ ( X ) = − π 2 4/5 ℑ y 1 ( 2 7/5 X ) . (273) The last relation is precisely the scaling formula Eq. (30) of the Main Text. Thus the two critical observables are respectively the real and imaginary parts of the same Painlevé-I solution.
Allowed side: X → + ∞ X\to+\infty X → + ∞
For positive t t t y 1 ( t ) ∼ − i t / 6 y_1(t)\sim-i\sqrt{t/6} y 1 ( t ) ∼ − i t /6 t t t
y 1 ( t ) = − i 6 t 1 / 2 [ 1 + ∑ n ≥ 1 c n t − 5 n / 2 ] , c 0 = 1 , (274) y_1(t)=-\frac{i}{\sqrt{6}}t^{1/2}\left[1+\sum_{n\geq1}c_nt^{-5n/2}\right],\qquad c_0=1, \tag*{(274)} y 1 ( t ) = − 6 i t 1/2 [ 1 + n ≥ 1 ∑ c n t − 5 n /2 ] , c 0 = 1 , (274) into the Painlevé-I equation (S175) and matching successive powers of t − 5 / 2 t^{-5/2} t − 5/2
c n = i [ 25 ( n − 1 ) 2 − 1 ] 8 6 c n − 1 − 1 2 ∑ j = 1 n − 1 c j c n − j , n ≥ 1. (275) c_n=\frac{i[25(n-1)^2-1]}{8\sqrt{6}}c_{n-1}-\frac{1}{2}\sum_{j=1}^{n-1}c_jc_{n-j},\qquad n\geq1. \tag*{(275)} c n = 8 6 i [ 25 ( n − 1 ) 2 − 1 ] c n − 1 − 2 1 j = 1 ∑ n − 1 c j c n − j , n ≥ 1. (275) The first coefficients are
c 1 = − i 6 48 , c 2 = 49 768 , c 3 = 1225 i 6 9216 , c 4 = − 4412401 1179648 . (276) c_1=-\frac{i\sqrt{6}}{48},\qquad c_2=\frac{49}{768},\qquad c_3=\frac{1225i\sqrt{6}}{9216},\qquad c_4=-\frac{4412401}{1179648}. \tag*{(276)} c 1 = − 48 i 6 , c 2 = 768 49 , c 3 = 9216 1225 i 6 , c 4 = − 1179648 4412401 . (276) The recursion alternates between purely imaginary odd coefficients and real even coefficients. Through Eq. (273) , the even coefficients therefore generate the algebraic expansion of Φ \Phi Φ Ψ \Psi Ψ
Substituting t = 2 7 / 5 X t=2^{7/5}X t = 2 7/5 X
Φ ( X ) ∼ 4 3 π 3 X 3 / 2 + C Φ + ∑ m ≥ 1 A m X 3 / 2 − 5 m , A m = 2 2 − 7 m π 3 ( 3 − 10 m ) c 2 m , \Phi(X)\sim\frac{4}{3\pi\sqrt{3}}X^{3/2}+C_{\Phi}+\sum_{m\geq1}A_mX^{3/2-5m},\qquad A_m=\frac{2^{2-7m}}{\pi\sqrt{3}(3-10m)}c_{2m}, Φ ( X ) ∼ 3 π 3 4 X 3/2 + C Φ + m ≥ 1 ∑ A m X 3/2 − 5 m , A m = π 3 ( 3 − 10 m ) 2 2 − 7 m c 2 m , Ψ ( X ) ∼ ∑ m ≥ 0 B m X − 1 − 5 m , B m = − 2 − 2 − 7 m ( 1 + 5 m ) 6 Im c 2 m + 1 . (277) \Psi(X)\sim\sum_{m\geq0}B_mX^{-1-5m},\qquad B_m=-\frac{2^{-2-7m}}{(1+5m)\sqrt{6}}\operatorname{Im}c_{2m+1}. \tag*{(277)} Ψ ( X ) ∼ m ≥ 0 ∑ B m X − 1 − 5 m , B m = − ( 1 + 5 m ) 6 2 − 2 − 7 m Im c 2 m + 1 . (277) The local Painlevé expansion determines Ψ \Psi Ψ C Φ C_{\Phi} C Φ X X X
Φ ( X ) ∼ 4 3 π 3 X 3 / 2 + C Φ − 7 24576 π 3 X − 7 / 2 + 259553 4831838208 π 3 X − 17 / 2 + ⋯ , \Phi(X)\sim\frac{4}{3\pi\sqrt{3}}X^{3/2}+C_{\Phi}-\frac{7}{24576\pi\sqrt{3}}X^{-7/2}+\frac{259553}{4831838208\pi\sqrt{3}}X^{-17/2}+\cdots, Φ ( X ) ∼ 3 π 3 4 X 3/2 + C Φ − 24576 π 3 7 X − 7/2 + 4831838208 π 3 259553 X − 17/2 + ⋯ , Ψ ( X ) ∼ 1 192 X − 1225 28311552 X 6 + ⋯ , X → + ∞ . (278) \Psi(X)\sim\frac{1}{192X}-\frac{1225}{28311552X^6}+\cdots,\qquad X\to+\infty. \tag*{(278)} Ψ ( X ) ∼ 192 X 1 − 28311552 X 6 1225 + ⋯ , X → + ∞. (278) Differentiating the leading term of Φ \Phi Φ Φ ′ ( X ) ∼ 2 X 1 / 2 / ( π 3 ) \Phi'(X)\sim2X^{1/2}/(\pi\sqrt{3}) Φ ′ ( X ) ∼ 2 X 1/2 / ( π 3 ) (31) of the Main Text. The subleading terms in Eq. (278) are therefore the systematic Painlevé corrections to the sharp large-N N N
Forbidden side: X → − ∞ X\to-\infty X → − ∞
The opposite limit has a qualitatively different structure. Put x = − t = 2 7 / 5 ∣ X ∣ > 0 x=-t=2^{7/5}|X|>0 x = − t = 2 7/5 ∣ X ∣ > 0 (211) then reads d 2 y / d x 2 = 6 y 2 − x \mathrm{d}^2y/\mathrm{d}x^2=6y^2-x d 2 y / d x 2 = 6 y 2 − x y 1 ( − x ) ∼ x / 6 y_1(-x)\sim\sqrt{x/6} y 1 ( − x ) ∼ x /6
Algebraic, or zero-instanton, sector.— Write
y 1 ( 0 ) ( − x ) = x 6 [ 1 + ∑ n ≥ 1 a n x − 5 n / 2 ] , a 0 = 1. (279) y_1^{(0)}(-x)=\sqrt{\frac{x}{6}}\left[1+\sum_{n\geq1}a_nx^{-5n/2}\right],\qquad a_0=1. \tag*{(279)} y 1 ( 0 ) ( − x ) = 6 x [ 1 + n ≥ 1 ∑ a n x − 5 n /2 ] , a 0 = 1. (279) Substitution into Painlevé I and matching powers of x − 5 / 2 x^{-5/2} x − 5/2
a n = 25 ( n − 1 ) 2 − 1 8 6 a n − 1 − 1 2 ∑ j = 1 n − 1 a j a n − j , n ≥ 1. (280) a_n=\frac{25(n-1)^2-1}{8\sqrt{6}}a_{n-1}-\frac{1}{2}\sum_{j=1}^{n-1}a_ja_{n-j},\qquad n\geq1. \tag*{(280)} a n = 8 6 25 ( n − 1 ) 2 − 1 a n − 1 − 2 1 j = 1 ∑ n − 1 a j a n − j , n ≥ 1. (280) The first coefficients are
a 1 = − 6 48 , a 2 = − 49 768 , a 3 = − 1225 6 9216 , a 4 = − 4412401 1179648 , (281) a_1=-\frac{\sqrt{6}}{48},\qquad a_2=-\frac{49}{768},\qquad a_3=-\frac{1225\sqrt{6}}{9216},\qquad a_4=-\frac{4412401}{1179648}, \tag*{(281)} a 1 = − 48 6 , a 2 = − 768 49 , a 3 = − 9216 1225 6 , a 4 = − 1179648 4412401 , (281) and hence
y 1 ( 0 ) ( − x ) = x 6 − 1 48 x − 2 − 49 768 6 x − 9 / 2 + ⋯ . (282) y_1^{(0)}(-x)=\sqrt{\frac{x}{6}}-\frac{1}{48}x^{-2}-\frac{49}{768\sqrt{6}}x^{-9/2}+\cdots. \tag*{(282)} y 1 ( 0 ) ( − x ) = 6 x − 48 1 x − 2 − 768 6 49 x − 9/2 + ⋯ . (282) Every term in this expansion is real. Equation (273) therefore implies that Φ ( X ) \Phi(X) Φ ( X ) X → − ∞ X\to-\infty X → − ∞
One-instanton fluctuation.— To determine the leading exponentially small contribution, linearize around the algebraic solution by writing y 1 ( − x ) = y 1 ( 0 ) ( − x ) + δ y ( x ) y_1(-x)=y_1^{(0)}(-x)+\delta y(x) y 1 ( − x ) = y 1 ( 0 ) ( − x ) + δ y ( x ) δ y ′ ′ = 12 y 1 ( 0 ) ( − x ) δ y \delta y''=12y_1^{(0)}(-x)\delta y δ y ′′ = 12 y 1 ( 0 ) ( − x ) δ y δ y ( x ) ∼ A ( x ) e − S ( x ) \delta y(x) \sim A(x)e^{-S(x)} δ y ( x ) ∼ A ( x ) e − S ( x ) A ( x ) A(x) A ( x ) x x x [ S ′ ( x ) ] 2 ≃ 12 y 1 ( 0 ) ( − x ) [S'(x)]^2 \simeq12y_1^{(0)}(-x) [ S ′ ( x ) ] 2 ≃ 12 y 1 ( 0 ) ( − x ) 12 y 1 ( 0 ) ( − x ) ∼ 2 6 x 1 / 2 12y_1^{(0)}(-x) \sim2\sqrt{6}x^{1/2} 12 y 1 ( 0 ) ( − x ) ∼ 2 6 x 1/2 S ′ ( x ) ∼ κ x 1 / 4 S'(x) \sim\kappa x^{1/4} S ′ ( x ) ∼ κ x 1/4 κ = 2 6 = 2 3 / 4 3 1 / 4 \kappa= \sqrt{2\sqrt{6}} = 2^{3/4}3^{1/4} κ = 2 6 = 2 3/4 3 1/4
S ( x ) = ∫ x κ u 1 / 4 d u = 4 κ 5 x 5 / 4 = 2 11 / 4 3 1 / 4 5 x 5 / 4 . (283) S(x)=\int^x \kappa u^{1/4}\,\mathrm{d}u=\frac{4\kappa}{5}x^{5/4}=\frac{2^{11/4}3^{1/4}}{5}x^{5/4}. \tag*{(283)} S ( x ) = ∫ x κ u 1/4 d u = 5 4 κ x 5/4 = 5 2 11/4 3 1/4 x 5/4 . (283) The irrelevant additive constant in S S S δ y \delta y δ y S ( x ) S(x) S ( x ) δ y ∝ e − S ( x ) \delta y \propto e^{-S(x)} δ y ∝ e − S ( x )
y ( 1 ) ( x ) = x − 1 / 8 e − S ( x ) ∑ n ≥ 0 b n x − 5 n / 4 , b 0 = 1. (284) y^{(1)}(x)=x^{-1/8}e^{-S(x)}\sum_{n\geq0}b_nx^{-5n/4},\qquad b_0=1. \tag*{(284)} y ( 1 ) ( x ) = x − 1/8 e − S ( x ) n ≥ 0 ∑ b n x − 5 n /4 , b 0 = 1. (284) The powers now occur in steps of x − 5 / 4 x^{-5/4} x − 5/4 ρ n = − 1 / 8 − 5 n / 4 \rho_n=-1/8-5n/4 ρ n = − 1/8 − 5 n /4 (284) into the linearized equation gives
b n = 2 5 κ n [ κ 2 ∑ k = 1 ⌊ ( n + 1 ) / 2 ⌋ a k b n + 1 − 2 k − ρ n − 1 ( ρ n − 1 − 1 ) b n − 1 ] , n ≥ 1. (285) b_n=\frac{2}{5\kappa n}\left[\kappa^2\sum_{k=1}^{\lfloor(n+1)/2\rfloor}a_kb_{n+1-2k}-\rho_{n-1}(\rho_{n-1}-1)b_{n-1}\right],\qquad n\geq1. \tag*{(285)} b n = 5 κn 2 κ 2 k = 1 ∑ ⌊( n + 1 ) /2 ⌋ a k b n + 1 − 2 k − ρ n − 1 ( ρ n − 1 − 1 ) b n − 1 , n ≥ 1. (285) The first coefficients are
b 1 = − 5 32 κ , b 2 = 75 6 8192 . (286) b_1=-\frac{5}{32\kappa},\qquad b_2=\frac{75\sqrt{6}}{8192}. \tag*{(286)} b 1 = − 32 κ 5 , b 2 = 8192 75 6 . (286) Thus, once the perturbative coefficients a n a_n a n
Kapaev Stokes multiplier and the physical branch. —The linearized analysis above determines the shape of the one-instanton sector but not its overall normalization. This normalization is fixed by the Stokes data of the particular tritronquée selected by the outgoing contour.
We use here the quasi-linear Stokes formula of Kapaev [59 ] . His Painlevé-I equation has exactly our canonical normalization, y ′ ′ ( ζ ) = 6 y ( ζ ) 2 + ζ y''(\zeta)=6y(\zeta)^2+\zeta y ′′ ( ζ ) = 6 y ( ζ ) 2 + ζ y 1 y_1 y 1 y 0 K y_0^{\mathrm K} y 0 K y 1 K y_1^{\mathrm K} y 1 K [59 ] . Corollary 2.4, Eq. (2.67) of that reference, gives their exponentially small difference in the sector containing the negative real axis:
y 1 K ( ζ ) − y 0 K ( ζ ) ∼ i π 2 − 11 / 8 3 − 1 / 8 ( e − i π ζ ) − 1 / 8 exp [ − 2 11 / 4 3 1 / 4 5 ( e − i π ζ ) 5 / 4 ] . (287) y_1^{\mathrm K}(\zeta)-y_0^{\mathrm K}(\zeta)\sim\frac{\mathrm i}{\sqrt{\pi}}2^{-11/8}3^{-1/8}(e^{-\mathrm i\pi}\zeta)^{-1/8}\exp\left[-\frac{2^{11/4}3^{1/4}}{5}(e^{-\mathrm i\pi}\zeta)^{5/4}\right]. \tag*{(287)} y 1 K ( ζ ) − y 0 K ( ζ ) ∼ π i 2 − 11/8 3 − 1/8 ( e − i π ζ ) − 1/8 exp [ − 5 2 11/4 3 1/4 ( e − i π ζ ) 5/4 ] . (287) In Eq. (287) we use only the leading exponentially small term of Kapaev’s Stokes formula. This term fixes the overall normalization of the one-instanton sector; the algebraic corrections multiplying the same exponential are generated independently by the recursion (S249).
The dictionary with our variables is simply ζ = t = − x \zeta=t=-x ζ = t = − x x > 0 x>0 x > 0 e − i π ζ = x e^{-\mathrm i\pi}\zeta=x e − i π ζ = x (287) is precisely e − S ( x ) e^{-S(x)} e − S ( x ) S ( x ) S(x) S ( x ) (283) . Thus Kapaev’s result becomes
y 1 K ( − x ) − y 0 K ( − x ) ∼ i π 2 − 11 / 8 3 − 1 / 8 x − 1 / 8 e − S ( x ) . (288) y_1^{\mathrm K}(-x)-y_0^{\mathrm K}(-x)\sim\frac{\mathrm i}{\sqrt{\pi}}2^{-11/8}3^{-1/8}x^{-1/8}e^{-S(x)}. \tag*{(288)} y 1 K ( − x ) − y 0 K ( − x ) ∼ π i 2 − 11/8 3 − 1/8 x − 1/8 e − S ( x ) . (288) On this Stokes ray the two lateral tritronquée solutions are complex conjugates and have the same real algebraic expansion. Their imaginary parts are therefore one half of the jump in Eq. (288) , with opposite signs. The physical E + i 0 E+\mathrm i0 E + i 0 (231) is the branch with negative imaginary part: indeed Eq. (273) requires Im y 1 < 0 \operatorname{Im}y_1<0 Im y 1 < 0
Im y 1 ( − x ) ∼ − 2 − 19 / 8 3 − 1 / 8 π x − 1 / 8 e − S ( x ) [ 1 + ∑ n ≥ 1 b n x − 5 n / 4 ] . (289) \operatorname{Im}y_1(-x)\sim-\frac{2^{-19/8}3^{-1/8}}{\sqrt{\pi}}x^{-1/8}e^{-S(x)}\left[1+\sum_{n\geq1}b_nx^{-5n/4}\right]. \tag*{(289)} Im y 1 ( − x ) ∼ − π 2 − 19/8 3 − 1/8 x − 1/8 e − S ( x ) [ 1 + n ≥ 1 ∑ b n x − 5 n /4 ] . (289) The overall factor in this equation is fixed by Kapaev’s Stokes formula, while the coefficients b n b_n b n
Critical IDOS tail. — Since x = 2 7 / 5 ∣ X ∣ x=2^{7/5}|X| x = 2 7/5 ∣ X ∣
S ( x ) = B ∣ X ∣ 5 / 4 , B : = 16 5 2 3 1 / 4 . (290) S(x)=B|X|^{5/4},\qquad B:=\frac{16}{5}\sqrt{2}\,3^{1/4}. \tag*{(290)} S ( x ) = B ∣ X ∣ 5/4 , B := 5 16 2 3 1/4 . (290) Using Eq. (273) , one obtains
Φ ′ ( X ) ∼ K ∣ X ∣ − 1 / 8 e − B ∣ X ∣ 5 / 4 [ 1 + ∑ n ≥ 1 q n ∣ X ∣ − 5 n / 4 ] , K = 2 − 7 / 4 3 − 1 / 8 π 3 / 2 , q n = 2 − 7 n / 4 b n . (291) \Phi'(X)\sim K|X|^{-1/8}e^{-B|X|^{5/4}}\left[1+\sum_{n\geq1}q_n|X|^{-5n/4}\right],\qquad K=\frac{2^{-7/4}3^{-1/8}}{\pi^{3/2}},\qquad q_n=2^{-7n/4}b_n. \tag*{(291)} Φ ′ ( X ) ∼ K ∣ X ∣ − 1/8 e − B ∣ X ∣ 5/4 [ 1 + n ≥ 1 ∑ q n ∣ X ∣ − 5 n /4 ] , K = π 3/2 2 − 7/4 3 − 1/8 , q n = 2 − 7 n /4 b n . (291) Together with the physical normalization Φ ( − ∞ ) = 0 \Phi(-\infty)=0 Φ ( − ∞ ) = 0
Φ ( X ) ∼ C Φ ( − ) ∣ X ∣ − 3 / 8 e − B ∣ X ∣ 5 / 4 [ 1 + ∑ n ≥ 1 p n ∣ X ∣ − 5 n / 4 ] , C Φ ( − ) = 1 2 17 / 4 3 3 / 8 π 3 / 2 . (292) \Phi(X)\sim C_\Phi^{(-)}|X|^{-3/8}e^{-B|X|^{5/4}}\left[1+\sum_{n\geq1}p_n|X|^{-5n/4}\right],\qquad C_\Phi^{(-)}=\frac{1}{2^{17/4}3^{3/8}\pi^{3/2}}. \tag*{(292)} Φ ( X ) ∼ C Φ ( − ) ∣ X ∣ − 3/8 e − B ∣ X ∣ 5/4 [ 1 + n ≥ 1 ∑ p n ∣ X ∣ − 5 n /4 ] , C Φ ( − ) = 2 17/4 3 3/8 π 3/2 1 . (292) The coefficients p n p_n p n (291) . Defining β = 4 2 3 1 / 4 \beta=4\sqrt{2}\,3^{1/4} β = 4 2 3 1/4 p 0 = 1 p_0=1 p 0 = 1
p n = q n − 10 n − 7 8 β p n − 1 , n ≥ 1 , p 1 = − 17 2 3 3 / 4 768 . (293) p_n=q_n-\frac{10n-7}{8\beta}p_{n-1},\qquad n\geq1,\qquad p_1=-\frac{17\sqrt{2}\,3^{3/4}}{768}. \tag*{(293)} p n = q n − 8 β 10 n − 7 p n − 1 , n ≥ 1 , p 1 = − 768 17 2 3 3/4 . (293) The first refined forbidden-side asymptotics is therefore
Φ ( X ) ∼ ∣ X ∣ − 3 / 8 2 17 / 4 3 3 / 8 π 3 / 2 exp [ − 16 5 2 3 1 / 4 ∣ X ∣ 5 / 4 ] [ 1 − 17 2 3 3 / 4 768 ∣ X ∣ − 5 / 4 + ⋯ ] . (294) \Phi(X)\sim\frac{|X|^{-3/8}}{2^{17/4}3^{3/8}\pi^{3/2}}\exp\left[-\frac{16}{5}\sqrt{2}\,3^{1/4}|X|^{5/4}\right]\left[1-\frac{17\sqrt{2}\,3^{3/4}}{768}|X|^{-5/4}+\cdots\right]. \tag*{(294)} Φ ( X ) ∼ 2 17/4 3 3/8 π 3/2 ∣ X ∣ − 3/8 exp [ − 5 16 2 3 1/4 ∣ X ∣ 5/4 ] [ 1 − 768 17 2 3 3/4 ∣ X ∣ − 5/4 + ⋯ ] . (294) Taking the logarithm of Eq. (291) gives at leading order
log Φ ′ ( X ) ∼ − 16 5 2 3 1 / 4 ∣ X ∣ 5 / 4 , X → − ∞ , (295) \log\Phi'(X)\sim-\frac{16}{5}\sqrt{2}\,3^{1/4}|X|^{5/4},\qquad X\to-\infty, \tag*{(295)} log Φ ′ ( X ) ∼ − 5 16 2 3 1/4 ∣ X ∣ 5/4 , X → − ∞ , (295) which is Eq. (32) of the Main Text.
In physical variables, ∣ X ∣ = N 4 / 5 ( E ∗ − E ) / s J 2 |X|=N^{4/5}(E^*-E)/s_J^2 ∣ X ∣ = N 4/5 ( E ∗ − E ) / s J 2 B / T B/T B / T B = 4 5 2 3 1 / 4 s J 1 / 2 ( E ∗ − E ) 5 / 4 B=\frac{4}{5}\sqrt{2}\,3^{1/4}s_J^{1/2}(E^*-E)^{5/4} B = 5 4 2 3 1/4 s J 1/2 ( E ∗ − E ) 5/4 [30 ] . Its deep forbidden-tail limit gives 4 ∣ E ∣ 3 / 2 / ( 3 T ) 4|E|^{3/2}/(3T) 4∣ E ∣ 3/2 / ( 3 T ) N N N (28) of the Main Text and in Sec. VIII C above.
Multi-instanton structure. — The complete forbidden-side solution is naturally organized as a Painlevé trans-series,
y 1 ( − x ) = y 1 ( 0 ) ( − x ) + ∑ k ≥ 1 σ k y ( k ) ( x ) , y ( k ) ( x ) ∼ x ( 4 − 5 k ) / 8 e − k S ( x ) [ series in x − 5 / 4 ] . (296) y_1(-x)=y_1^{(0)}(-x)+\sum_{k\geq1}\sigma^k y^{(k)}(x),\qquad y^{(k)}(x)\sim x^{(4-5k)/8}e^{-kS(x)}\left[\text{series in }x^{-5/4}\right]. \tag*{(296)} y 1 ( − x ) = y 1 ( 0 ) ( − x ) + k ≥ 1 ∑ σ k y ( k ) ( x ) , y ( k ) ( x ) ∼ x ( 4 − 5 k ) /8 e − k S ( x ) [ series in x − 5/4 ] . (296) For the physical tritronquée the parameter σ \sigma σ y ( 1 ) , … , y ( k − 1 ) y^{(1)},\ldots,y^{(k-1)} y ( 1 ) , … , y ( k − 1 ) y ( k ) y^{(k)} y ( k )
On the physical Stokes line σ \sigma σ y 1 y_1 y 1 (273) that the IDOS contains the sectors e − S e^{-S} e − S e − 3 S e^{-3S} e − 3 S e − 5 S e^{-5S} e − 5 S … \ldots … e − 2 S e^{-2S} e − 2 S e − 4 S e^{-4S} e − 4 S … \ldots … N N N
Forbidden-side Lyapunov expansion. — The algebraic part of Ψ \Psi Ψ (279) into Eq. (273) gives
Ψ ′ ( X ) ∼ 2 3 ∣ X ∣ 1 / 2 [ 1 + ∑ n ≥ 1 a n 2 − 7 n / 2 ∣ X ∣ − 5 n / 2 ] . (297) \Psi'(X)\sim\frac{2}{\sqrt{3}}|X|^{1/2}\left[1+\sum_{n\geq1}a_n2^{-7n/2}|X|^{-5n/2}\right]. \tag*{(297)} Ψ ′ ( X ) ∼ 3 2 ∣ X ∣ 1/2 [ 1 + n ≥ 1 ∑ a n 2 − 7 n /2 ∣ X ∣ − 5 n /2 ] . (297) Term-by-term integration yields
Ψ ( X ) ∼ C Ψ ( − ) − 4 3 3 ∣ X ∣ 3 / 2 + ∑ n ≥ 1 D n ∣ X ∣ ( 3 − 5 n ) / 2 , D n = − 4 2 − 7 n / 2 3 ( 3 − 5 n ) a n . (298) \Psi(X)\sim C_\Psi^{(-)}-\frac{4}{3\sqrt{3}}|X|^{3/2}+\sum_{n\geq1}D_n|X|^{(3-5n)/2},\qquad D_n=-\frac{4\,2^{-7n/2}}{\sqrt{3}(3-5n)}a_n. \tag*{(298)} Ψ ( X ) ∼ C Ψ ( − ) − 3 3 4 ∣ X ∣ 3/2 + n ≥ 1 ∑ D n ∣ X ∣ ( 3 − 5 n ) /2 , D n = − 3 ( 3 − 5 n ) 4 2 − 7 n /2 a n . (298) The first terms are
Ψ ( X ) ∼ C Ψ ( − ) − 4 3 3 ∣ X ∣ 3 / 2 − 1 192 ∣ X ∣ − 7 3 73728 ∣ X ∣ − 7 / 2 − 1225 28311552 ∣ X ∣ − 6 + ⋯ . (299) \Psi(X) \sim C_{\Psi}^{(-)} - \frac{4}{3\sqrt{3}} |X|^{3/2} - \frac{1}{192|X|} - \frac{7\sqrt{3}}{73728}|X|^{-7/2} - \frac{1225}{28311552}|X|^{-6} + \cdots. \tag*{(299)} Ψ ( X ) ∼ C Ψ ( − ) − 3 3 4 ∣ X ∣ 3/2 − 192∣ X ∣ 1 − 73728 7 3 ∣ X ∣ − 7/2 − 28311552 1225 ∣ X ∣ − 6 + ⋯ . (299) The additive constant C Ψ ( − ) C_{\Psi}^{(-)} C Ψ ( − )
The resulting hierarchy is therefore completely systematic. The zero-instanton sector generates the algebraic expansion of the Lyapunov scaling function; the one-instanton sector, together with the Kapaev Stokes multiplier, gives the full leading Lifshitz-tail expansion of the IDOS; the two-instanton sector gives the first exponentially small correction to Ψ \Psi Ψ Φ \Phi Φ
Global normalization and numerical evaluation of the critical scaling function
The local Painlevé asymptotics of Sec. IX determine Φ ′ ( X ) \Phi'(X) Φ ′ ( X ) C Φ C_{\Phi} C Φ Φ ( − ∞ ) = 0 \Phi(-\infty)=0 Φ ( − ∞ ) = 0
Using Eq. (S237), the physical normalization gives
Φ ( X ) = − 2 − 3 / 5 π ∫ − ∞ 7 X / 5 Im y 1 ( t ) d t . (300) \Phi(X) = -\frac{2^{-3/5}}{\pi}\int_{-\infty}^{7X/5} \operatorname{Im} y_1(t)\,\mathrm{d}t. \tag*{(300)} Φ ( X ) = − π 2 − 3/5 ∫ − ∞ 7 X /5 Im y 1 ( t ) d t . (300) Thus the problem is genuinely global: the same solution y 1 y_1 y 1 t → − ∞ t \to-\infty t → − ∞ t → + ∞ t \to+\infty t → + ∞ y 1 ( t ) y_1(t) y 1 ( t )
Global boundary-value problem.— A shooting method would integrate the Painlevé-I equation from initial values of y 1 y_1 y 1 y 1 ′ y_1' y 1 ′ t t t t t t
We restrict the numerical computation to the finite interval − L ≤ t ≤ R -L \le t \le R − L ≤ t ≤ R L = 10 L=10 L = 10 R = 30 R=30 R = 30 t = − L t=-L t = − L a 6 a_6 a 6 b 7 b_7 b 7 t = R t=R t = R c 7 c_7 c 7
In the adaptive collocation method, y 1 ( t ) y_1(t) y 1 ( t ) 5 × 10 − 11 5 \times10^{-11} 5 × 1 0 − 11 L L L R R R
As a representative value,
y 1 ( 0 ) = 0.1517346228157 − 0.1102416564465 i . (301) y_1(0) = 0.1517346228157 - 0.1102416564465\,\mathrm{i}. \tag*{(301)} y 1 ( 0 ) = 0.1517346228157 − 0.1102416564465 i . (301) Global normalization and determination of C Φ C_{\Phi} C Φ Once y 1 y_1 y 1 Φ \Phi Φ ( − ∞ , − L ) (-\infty,-L) ( − ∞ , − L )
Φ ( 0 ) = 0.01191950433362. (302) \Phi(0) = 0.01191950433362. \tag*{(302)} Φ ( 0 ) = 0.01191950433362. (302) The global constant is extracted by subtracting the known allowed-side asymptotic terms. Define
C Φ ( X ) : = Φ ( X ) − 4 3 π 3 X 3 / 2 + 7 24576 π 3 X − 7 / 2 − 259553 4831838208 π 3 X − 17 / 2 . (303) C_{\Phi}(X) := \Phi(X) - \frac{4}{3\pi\sqrt{3}}X^{3/2} + \frac{7}{24576\pi\sqrt{3}}X^{-7/2}
- \frac{259553}{4831838208\pi\sqrt{3}}X^{-17/2}. \tag*{(303)} C Φ ( X ) := Φ ( X ) − 3 π 3 4 X 3/2 + 24576 π 3 7 X − 7/2 − 4831838208 π 3 259553 X − 17/2 . (303) According to Eq. (278) , C Φ = lim X → + ∞ C Φ ( X ) C_{\Phi}=\lim_{X\to+\infty}C_{\Phi}(X) C Φ = lim X → + ∞ C Φ ( X )
C Φ n u m = O ( 10 − 13 ) , i.e. C Φ = 0 within numerical accuracy . (304) C_{\Phi}^{\mathrm{num}}=O(10^{-13}), \qquad\text{i.e.}\qquad C_{\Phi}=0\quad\text{within numerical accuracy}. \tag*{(304)} C Φ num = O ( 1 0 − 13 ) , i.e. C Φ = 0 within numerical accuracy . (304) This is the main global result of the calculation. Neither local asymptotic expansion determines C Φ C_{\Phi} C Φ Φ ( − ∞ ) = 0 \Phi(-\infty)=0 Φ ( − ∞ ) = 0
The resulting scaling function is shown in Figure S2 . It is exponentially small on the forbidden side and rapidly approaches the planar X 3 / 2 X^{3/2} X 3/2
Figure S2. Critical IDOS scaling function Φ ( X ) \Phi(X) Φ ( X ) y 1 y_1 y 1 C Φ = 0 C_{\Phi}=0 C Φ = 0 X X X
Allowed-side asymptotic check.— With the globally determined value C Φ = 0 C_{\Phi}=0 C Φ = 0 t = 2 7 / 5 X t=2^{7/5}X t = 2 7/5 X
Table S269.
Thus the large-X X X X ≃ 2 X\simeq2 X ≃ 2
Forbidden-side asymptotic check.— On the opposite side the IDOS is the one-instanton Stokes contribution derived in Eq. (294) . Its relative errors against the global numerical solution are
Table S270.
The numerical and one-instanton curves are essentially indistinguishable on a logarithmic scale for X ≲ − 1.5 X\lesssim-1.5 X ≲ − 1.5 Figure S3 .
Figure S3. Forbidden-side critical IDOS. The solid line is the global Painlevé-I result and the dashed line is the one-instanton expansion (S258), including its first algebraic correction. The agreement rapidly improves as X → − ∞ X \to-\infty X → − ∞
The two asymptotic comparisons probe complementary parts of the construction. The allowed side tests the ordinary Painlevé-I power series and the metric factor t = 2 7 / 5 X t = 2^{7/5}X t = 2 7/5 X Φ ( − ∞ ) = 0 \Phi(-\infty) = 0 Φ ( − ∞ ) = 0
For reference, numerical values of the globally normalized scaling function are collected in Table S1 .
Table S1. Numerical values of the globally normalized critical IDOS scaling function Φ ( X ) \Phi(X) Φ ( X )
Comparison with the localized random-band edge
We now compare the critical function Φ ( X ) \Phi(X) Φ ( X )
Historical context and Sodin’s localized edge
Random band matrices interpolate between mean-field random matrices and quasi-one-dimensional localization. The early works [60 , 61 , 62 ] established the characteristic bandwidth-dependent localization-volume scaling and its nonlinear-σ \sigma σ
The spectral edge was analyzed diagrammatically by Silvestrov [65 ] , who used the topological organization of band-matrix graphs and identified the non-mean-field edge crossover. Sodin [66 ] subsequently gave a rigorous treatment for M × M M \times M M × M W W W M M M W ∼ M 5 / 6 W \sim M^{5/6} W ∼ M 5/6 1 ≪ W ≪ M 5 / 6 1 \ll W \ll M^{5/6} 1 ≪ W ≪ M 5/6 W − 4 / 5 W^{-4/5} W − 4/5
M / W 6 / 5 M/W^{6/5} M / W 6/5 ξ = ( 2 − λ ) W 4 / 5 \xi= (2-\lambda)W^{4/5} ξ = ( 2 − λ ) W 4/5
We use the conventional normalization in which the limiting semicircle has upper edge 2. If λ α \lambda_\alpha λ α ξ α = ( 2 − λ α ) W 4 / 5 \xi_\alpha= (2-\lambda_\alpha)W^{4/5} ξ α = ( 2 − λ α ) W 4/5
σ 2 ( ξ ) : = lim M , W → ∞ 1 ≪ W ≪ M 5 / 6 W 6 / 5 M ⟨ # { α : ξ α < ξ } ⟩ . (305) \sigma_2(\xi) := \lim_{\substack{M,W\to\infty\\ 1\ll W\ll M^{5/6}}} \frac{W^{6/5}}{M}\left\langle\#\{\alpha:\xi_\alpha<\xi\}\right\rangle. \tag*{(305)} σ 2 ( ξ ) := M , W → ∞ 1 ≪ W ≪ M 5/6 lim M W 6/5 ⟨ # { α : ξ α < ξ } ⟩ . (305) This is the unitary, or random-phase, measure σ 2 \sigma_2 σ 2 [66 ] , after rescaling Sodin’s semicircle edge from 1 to 2.
Sodin’s Theorem 1.2 proves, in particular,
σ 2 ( ξ ) = 2 3 π ξ 3 / 2 + O ( ξ ) , ξ → + ∞ , σ 2 ( ξ ) ≤ C exp [ − c 0 ∣ ξ ∣ 5 / 4 ] , ξ → − ∞ , (306) \sigma_2(\xi)=\frac{2}{3\pi}\xi^{3/2}+O(\xi),\qquad\xi\to+\infty,\qquad\sigma_2(\xi)\le C\exp\left[-c_0|\xi|^{5/4}\right],\qquad\xi\to-\infty, \tag*{(306)} σ 2 ( ξ ) = 3 π 2 ξ 3/2 + O ( ξ ) , ξ → + ∞ , σ 2 ( ξ ) ≤ C exp [ − c 0 ∣ ξ ∣ 5/4 ] , ξ → − ∞ , (306) for positive constants C , c 0 C,c_0 C , c 0 W − 4 / 5 W^{-4/5} W − 4/5 W 6 / 5 W^{6/5} W 6/5 ∣ ξ ∣ 5 / 4 |\xi|^{5/4} ∣ ξ ∣ 5/4
Liu and Zou [67 ] subsequently developed the edge problem for more general band profiles and dimensions. Their Theorem 1.3, Eqs. (1.11)–(1.12), describes the subcritical regime, while Theorem 1.4, Eqs. (1.13)–(1.14), describes the critical regime. Let H H H M × M M\times M M × M H H H ⟨ n − 1 Tr U n ( H / 2 ) ⟩ \langle n^{-1}\operatorname{Tr}U_n(H/2)\rangle ⟨ n − 1 Tr U n ( H /2 )⟩ U n U_n U n n n n § 0.16 C. Near the upper edge, writing λ = 2 − δ \lambda=2-\delta λ = 2 − δ [67 ] depends on the combination n δ n\sqrt{\delta} n δ n n n δ ∼ n − 2 \delta\sim n^{-2} δ ∼ n − 2 n ∼ W 2 / 5 n\sim W^{2/5} n ∼ W 2/5 δ ∼ W − 4 / 5 \delta\sim W^{-4/5} δ ∼ W − 4/5 ϕ 3 \phi^3 ϕ 3 [68 ] established Markov-chain comparison principles, notably Theorems 1.3 and 1.4 there, which extend critical and subcritical edge statistics across broad classes of inhomogeneous variance profiles. These results strengthen the universality picture on the RBM side, but do not imply the correspondence with the Riccati scaling function proposed below. The recent works Refs. [63 , 64 ] concern broader progress on RBM delocalization and universality and are not used in the specific coefficient matching that follows.
The comparison requires three pieces of information already derived above. From Eq. (278) ,
Φ ( X ) ∼ A + X 3 / 2 + C Φ + a Φ X − 7 / 2 + O ( X − 17 / 2 ) , A + = 4 3 π 3 , a Φ = − 7 24576 π 3 , (307) \Phi(X) \sim A_{+}X^{3/2} + C_{\Phi} + a_{\Phi}X^{-7/2} + O(X^{-17/2}), \qquad A_{+} = \frac{4}{3\pi\sqrt{3}}, \qquad a_{\Phi} = -\frac{7}{24576\pi\sqrt{3}}, \tag*{(307)} Φ ( X ) ∼ A + X 3/2 + C Φ + a Φ X − 7/2 + O ( X − 17/2 ) , A + = 3 π 3 4 , a Φ = − 24576 π 3 7 , (307) as X → + ∞ X \to+\infty X → + ∞ § 0.15 gives C Φ = 0 C_{\Phi}=0 C Φ = 0 (304) ; we use this physical normalization in the regularized quantities below. On the forbidden side, Eq. (294) gives
log Φ ( X ) ∼ − A − ∣ X ∣ 5 / 4 , A − = 16 5 2 3 1 / 4 , X → − ∞ . (308) \log\Phi(X) \sim-A_{-}|X|^{5/4}, \qquad A_{-} = \frac{16}{5}\sqrt{2}3^{1/4}, \qquad X \to-\infty. \tag*{(308)} log Φ ( X ) ∼ − A − ∣ X ∣ 5/4 , A − = 5 16 2 3 1/4 , X → − ∞. (308) A second independent characteristic of the crossover is obtained by subtracting the sharp planar edge. Set X + = max ( X , 0 ) X_{+}=\max(X,0) X + = max ( X , 0 ) Φ 0 ( X ) = A + X + 3 / 2 \Phi_{0}(X)=A_{+}X_{+}^{3/2} Φ 0 ( X ) = A + X + 3/2 δ Φ ( X ) = Φ ( X ) − Φ 0 ( X ) \delta\Phi(X)=\Phi(X)-\Phi_{0}(X) δ Φ ( X ) = Φ ( X ) − Φ 0 ( X ) C Φ = 0 C_{\Phi}=0 C Φ = 0 δ Φ \delta\Phi δ Φ
b Φ : = 2 3 ∫ − ∞ ∞ δ Φ ( X ) d X . (309) b_{\Phi}:=\frac{2}{3}\int_{-\infty}^{\infty}\delta\Phi(X)\,\mathrm{d}X. \tag*{(309)} b Φ := 3 2 ∫ − ∞ ∞ δ Φ ( X ) d X . (309) With this normalization the moment can be obtained analytically, without numerical quadrature. After subtracting the same planar background, the critical form of the Thouless relation gives
δ Ψ ( X ) = ∫ − ∞ ∞ log ∣ X − Y ∣ δ Φ ′ ( Y ) d Y = ∫ − ∞ ∞ δ Φ ( Y ) X − Y d Y . (310) \delta\Psi(X)=\int_{-\infty}^{\infty}\log\lvert X-Y\rvert\,\delta\Phi'(Y)\,\mathrm{d}Y=\int_{-\infty}^{\infty}\frac{\delta\Phi(Y)}{X-Y}\,\mathrm{d}Y. \tag*{(310)} δ Ψ ( X ) = ∫ − ∞ ∞ log ∣ X − Y ∣ δ Φ ′ ( Y ) d Y = ∫ − ∞ ∞ X − Y δ Φ ( Y ) d Y . (310) Here polynomial and X X X (272) . Expanding Eq. (310) for X → + ∞ X\to+\infty X → + ∞
δ Ψ ( X ) = 1 X ∫ − ∞ ∞ δ Φ ( Y ) d Y + O ( X − 2 ) . (311) \delta\Psi(X)=\frac{1}{X}\int_{-\infty}^{\infty}\delta\Phi(Y)\,\mathrm{d}Y+O(X^{-2}). \tag*{(311)} δ Ψ ( X ) = X 1 ∫ − ∞ ∞ δ Φ ( Y ) d Y + O ( X − 2 ) . (311) On the other hand, the direct Painlevé-I expansion Eq. (278) gives Ψ ( X ) = 1 / ( 192 X ) + O ( X − 6 ) \Psi(X)=1/(192X)+O(X^{-6}) Ψ ( X ) = 1/ ( 192 X ) + O ( X − 6 )
∫ − ∞ ∞ δ Φ ( X ) d X = 1 192 , b Φ = 1 288 . (312) \int_{-\infty}^{\infty}\delta\Phi(X)\,\mathrm{d}X=\frac{1}{192}, \qquad b_{\Phi}=\frac{1}{288}. \tag*{(312)} ∫ − ∞ ∞ δ Φ ( X ) d X = 192 1 , b Φ = 288 1 . (312) Thus, once the global normalization C Φ = 0 C_{\Phi}=0 C Φ = 0 b Φ = 1 / 288 b_{\Phi}=1/288 b Φ = 1/288
Liu–Zou ribbon diagrams and the RBM shape coefficient
We now extract from the Liu–Zou expansion the RBM quantity that will fix the second metric factor in the conjecture. The detailed graph construction and the explicit evaluation of the relevant integrals are given in § 0.16 F; here we summarize the main mechanism and the resulting coefficient.
As explained above, Liu and Zou [67 ] probe the spectrum through disorder averages of normalized traces of high-degree Chebyshev-type polynomials of the random band matrix H H H ϕ 3 \phi^{3} ϕ 3
For a connected ribbon graph G = ( V , E ) G=(V,E) G = ( V , E ) V V V E E E α e > 0 \alpha_{e}>0 α e > 0 e ∈ E e\in E e ∈ E
u G ( { α e } ) = ∑ T s p a n n i n g t r e e ∏ e ∉ T α e . (313) u_{G}(\{\alpha_{e}\})=\sum_{T\ \mathrm{spanning\ tree}}\prod_{e\notin T}\alpha_{e}. \tag*{(313)} u G ({ α e }) = T spanning tree ∑ e ∈ / T ∏ α e . (313) which is Eq. (3.5) of Ref. [67 ] . In one spatial dimension the corresponding graph weight contains U G − 1 / 2 \mathcal{U}_{G}^{-1/2} U G − 1/2 α \alpha α p α ( x ) = ( 2 π ς 2 α ) − 1 / 2 exp [ − x 2 / ( 2 ς 2 α ) ] p_{\alpha}(x)=(2\pi\varsigma^{2}\alpha)^{-1/2}\exp[-x^{2}/(2\varsigma^{2}\alpha)] p α ( x ) = ( 2 π ς 2 α ) − 1/2 exp [ − x 2 / ( 2 ς 2 α )] ς 2 \varsigma^{2} ς 2
The first contribution that is sensitive to the detailed shape of the localized edge is a genus-one one-point diagram. In the Liu–Zou one-point convention it consists of a marked tail attached to a subdivided theta graph. Removing the marked tail leaves a four-edge core with continuum lengths α 1 , … , α 4 \alpha_{1},\ldots,\alpha_{4} α 1 , … , α 4
I s u b ( α ) = 1 2 π ς 2 1 α 3 α 4 + ( α 1 + α 2 ) ( α 3 + α 4 ) . (314) I_{\mathrm{sub}}(\boldsymbol{\alpha})=\frac{1}{2\pi\varsigma^{2}}\frac{1}{\sqrt{\alpha_{3}\alpha_{4}+(\alpha_{1}+\alpha_{2})(\alpha_{3}+\alpha_{4})}}. \tag*{(314)} I sub ( α ) = 2 π ς 2 1 α 3 α 4 + ( α 1 + α 2 ) ( α 3 + α 4 ) 1 . (314) The marked tail fixes the total graph length through the polynomial degree. Eliminating it introduces a hyperplane Jacobian which cancels the corresponding lattice-density factor in the Liu–Zou continuum formula. After rescaling the four remaining edge lengths by α e = ( τ / 2 ) u e \alpha_{e}=(\tau/2)u_{e} α e = ( τ /2 ) u e τ \tau τ 1 / n 1/n 1/ n τ \tau τ F Γ ( τ ) F_{\Gamma}(\tau) F Γ ( τ )
F Γ ( τ ) = τ 2 16 π ς 2 J , J = 2 3 ∫ 0 1 ( 1 − z ) arcsin z 4 − 3 z d z = π 27 . (315) F_{\Gamma}(\tau)=\frac{\tau^{2}}{16\pi\varsigma^{2}}J,\qquad J=\frac{2}{3}\int_{0}^{1}(1-z)\arcsin\sqrt{\frac{z}{4-3z}}\,\mathrm{d}z=\frac{\pi}{27}. \tag*{(315)} F Γ ( τ ) = 16 π ς 2 τ 2 J , J = 3 2 ∫ 0 1 ( 1 − z ) arcsin 4 − 3 z z d z = 27 π . (315) Thus
F Γ ( τ ) = τ 2 432 ς 2 . (316) F_{\Gamma}(\tau)=\frac{\tau^{2}}{432\varsigma^{2}}. \tag*{(316)} F Γ ( τ ) = 432 ς 2 τ 2 . (316) The integral J = π / 27 J=\pi/27 J = π /27 § 0.16 F. For Sodin’s sharp-cutoff band profile the rescaled longitudinal step is uniform on [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] ς 2 = 1 / 3 \varsigma^{2}=1/3 ς 2 = 1/3 1 / 144 1/144 1/144
We now relate this graph coefficient to Sodin’s one-edge cumulative intensity σ 2 ( ξ ) \sigma_{2}(\xi) σ 2 ( ξ ) τ \tau τ [67 ] . Since the matrix trace receives equal contributions from the upper and lower spectral edges, the unitary one-point transform is
ϕ 1 ( τ ) = 2 ∫ − ∞ ∞ sin ( τ ξ ) τ ξ d σ 2 ( ξ ) , (317) \phi_{1}(\tau)=2\int_{-\infty}^{\infty}\frac{\sin(\tau\sqrt{\xi})}{\tau\sqrt{\xi}}\,\mathrm{d}\sigma_{2}(\xi), \tag*{(317)} ϕ 1 ( τ ) = 2 ∫ − ∞ ∞ τ ξ sin ( τ ξ ) d σ 2 ( ξ ) , (317) with analytic continuation of the kernel for ξ < 0 \xi<0 ξ < 0 σ 2 \sigma_{2} σ 2
Subtract the leading semiclassical edge law by defining ξ + = max ( ξ , 0 ) \xi_{+}=\max(\xi,0) ξ + = max ( ξ , 0 )
δ σ 2 ( ξ ) = σ 2 ( ξ ) − 2 3 π ξ + 3 / 2 , b R B M : = 2 3 ∫ − ∞ ∞ δ σ 2 ( ξ ) d ξ . (318) \delta\sigma_{2}(\xi)=\sigma_{2}(\xi)-\frac{2}{3\pi}\xi_{+}^{3/2},\qquad b_{\mathrm{RBM}}:=\frac{2}{3}\int_{-\infty}^{\infty}\delta\sigma_{2}(\xi)\,\mathrm{d}\xi. \tag*{(318)} δ σ 2 ( ξ ) = σ 2 ( ξ ) − 3 π 2 ξ + 3/2 , b RBM := 3 2 ∫ − ∞ ∞ δ σ 2 ( ξ ) d ξ . (318) Here b R B M b_{\mathrm{RBM}} b RBM b Φ b_{\Phi} b Φ (309) . Expanding the sinc kernel for small τ \tau τ
ϕ 1 r e g ( τ ) = τ 2 3 ∫ − ∞ ∞ δ σ 2 ( ξ ) d ξ + o ( τ 2 ) = b R B M 2 τ 2 + o ( τ 2 ) . (319) \phi_{1}^{\mathrm{reg}}(\tau)=\frac{\tau^{2}}{3}\int_{-\infty}^{\infty}\delta\sigma_{2}(\xi)\,\mathrm{d}\xi+o(\tau^{2})=\frac{b_{\mathrm{RBM}}}{2}\tau^{2}+o(\tau^{2}). \tag*{(319)} ϕ 1 reg ( τ ) = 3 τ 2 ∫ − ∞ ∞ δ σ 2 ( ξ ) d ξ + o ( τ 2 ) = 2 b RBM τ 2 + o ( τ 2 ) . (319) Comparison with Eq. (316) at ς 2 = 1 / 3 \varsigma^{2}=1/3 ς 2 = 1/3
b R B M = 1 72 . (320) b_{\mathrm{RBM}}=\frac{1}{72}. \tag*{(320)} b RBM = 72 1 . (320) This is the RBM shape coefficient used below to determine the metric factors. It is obtained by an explicit evaluation within the Liu–Zou diagrammatic framework, rather than being a coefficient quoted from Ref. [67 ] . The full derivation, including the theta and subdivided-theta graphs, the marked-tail Jacobian cancellation and the evaluation of J J J § 0.16 F.
Metric matching and an overdetermining coefficient test
We can now formulate the one-point universality conjecture precisely. Allowing independent horizontal and vertical metric factors, we write
Φ ( c ξ ) = A σ 2 ( ξ ) , c > 0 , A > 0. (321) \Phi(c\xi)=\mathcal{A}\sigma_2(\xi),\qquad c>0,\qquad\mathcal{A}>0. \tag*{(321)} Φ ( c ξ ) = A σ 2 ( ξ ) , c > 0 , A > 0. (321) Here c c c ξ \xi ξ X X X A \mathcal{A} A
The leading allowed-side laws, Φ ( X ) ∼ A + X 3 / 2 \Phi(X)\sim A_+X^{3/2} Φ ( X ) ∼ A + X 3/2 A + = 4 / ( 3 π 3 ) A_+=4/(3\pi\sqrt{3}) A + = 4/ ( 3 π 3 ) σ 2 ( ξ ) ∼ 2 ξ 3 / 2 / ( 3 π ) \sigma_2(\xi)\sim2\xi^{3/2}/(3\pi) σ 2 ( ξ ) ∼ 2 ξ 3/2 / ( 3 π )
A = 2 3 c 3 / 2 . (322) \mathcal{A}=\frac{2}{\sqrt{3}}c^{3/2}. \tag*{(322)} A = 3 2 c 3/2 . (322) This relation alone cannot determine both metric factors. The second condition is supplied by the regularized moments. Because the leading backgrounds match under Eq. (322) , subtracting them from Eq. (321) and changing variables X = c ξ X=c\xi X = c ξ
b Φ = A c b R B M . (323) b_{\Phi}=\mathcal{A}c b_{\mathrm{RBM}}. \tag*{(323)} b Φ = A c b RBM . (323) Using b Φ = 1 / 288 b_{\Phi}=1/288 b Φ = 1/288 b R B M = 1 / 72 b_{\mathrm{RBM}}=1/72 b RBM = 1/72
c = ( 3 64 ) 1 / 5 , A = 48 − 1 / 5 , A c = 1 4 . (324) c=\left(\frac{3}{64}\right)^{1/5},\qquad\mathcal{A}=48^{-1/5},\qquad\mathcal{A}c=\frac{1}{4}. \tag*{(324)} c = ( 64 3 ) 1/5 , A = 4 8 − 1/5 , A c = 4 1 . (324) Numerically, c = 0.5422358856 … c=0.5422358856\ldots c = 0.5422358856 … A = 0.4610539557 … \mathcal{A}=0.4610539557\ldots A = 0.4610539557 …
Once c c c A \mathcal{A} A
σ 2 ( ξ ) = 2 3 π ξ 3 / 2 + C σ + a R B M ξ − 7 / 2 + O ( ξ − 17 / 2 ) , a R B M = − 7 2304 π , ξ → + ∞ . (325) \sigma_2(\xi)=\frac{2}{3\pi}\xi^{3/2}+C_{\sigma}+a_{\mathrm{RBM}}\xi^{-7/2}+O(\xi^{-17/2}),\qquad a_{\mathrm{RBM}}=-\frac{7}{2304\pi},\qquad\xi\to+\infty. \tag*{(325)} σ 2 ( ξ ) = 3 π 2 ξ 3/2 + C σ + a RBM ξ − 7/2 + O ( ξ − 17/2 ) , a RBM = − 2304 π 7 , ξ → + ∞. (325) Here C σ C_{\sigma} C σ a R B M a_{\mathrm{RBM}} a RBM c c c A \mathcal{A} A
Writing Eq. (321) equivalently as Φ ( X ) = A σ 2 ( X / c ) \Phi(X)=\mathcal{A}\sigma_2(X/c) Φ ( X ) = A σ 2 ( X / c ) X − 7 / 2 X^{-7/2} X − 7/2
a Φ = A a R B M c 7 / 2 = − 7 24576 π 3 , (326) a_{\Phi}=\mathcal{A}a_{\mathrm{RBM}}c^{7/2}=-\frac{7}{24576\pi\sqrt{3}}, \tag*{(326)} a Φ = A a RBM c 7/2 = − 24576 π 3 7 , (326) in exact agreement with the independently obtained Painlevé-I coefficient in Eq. (307) . This is therefore the first analytic overdetermining test of Eq. (321) , rather than part of the metric matching.
Independent checks, predictions, and status of the conjecture
The normalization entering Eq. (320) has two checks independent of the Riccati comparison. First, the same ribbon-graph conventions in the supercritical regime reproduce the GUE/Airy one-point transform: the unitary tadpole vanishes and the first loopless contribution gives the known coefficient 1 / 384 1/384 1/384 W = 16 W=16 W = 16 48 48 48 τ \tau τ 0.0065 ( 8 ) 0.0065(8) 0.0065 ( 8 ) 1 / 144 = 0.006944 … 1/144=0.006944\ldots 1/144 = 0.006944 …
A separate numerical test was performed directly on cyclic complex-Hermitian band matrices with ( M , W ) = ( 1024 , 16 ) , ( 2048 , 24 ) , ( 2048 , 32 ) (M,W)=(1024,16),(2048,24),(2048,32) ( M , W ) = ( 1024 , 16 ) , ( 2048 , 24 ) , ( 2048 , 32 ) (324) , the three data sets collapse onto the same crossover shape. Over approximately 0.8 ≲ ξ ≲ 2 0.8\lesssim\xi\lesssim2 0.8 ≲ ξ ≲ 2 2 2 2 5 % 5\% 5% W W W W W W σ 2 , W ( ξ ) ≃ α W ξ 3 / 2 \sigma_{2,W}(\xi)\simeq\alpha_W\xi^{3/2} σ 2 , W ( ξ ) ≃ α W ξ 3/2
α W 2 / ( 3 π ) = 0.904 , 0.926 , 0.963 ( W = 16 , 24 , 32 ) . (327) \frac{\alpha_W}{2/(3\pi)}=0.904,\quad0.926,\quad0.963\qquad(W=16,24,32). \tag*{(327)} 2/ ( 3 π ) α W = 0.904 , 0.926 , 0.963 ( W = 16 , 24 , 32 ) . (327) Removing this independently measured normalization deficit brings the crossover curves to the predicted shape at the few-percent level. We regard this as supporting numerical evidence, not as a proof of Eq. (321) .
The conjecture also makes two sharp predictions not used in fixing the metric. First, since the global calculation of § 0.15 gives C Φ = 0 C_{\Phi}=0 C Φ = 0 C Φ = A C σ C_{\Phi}=A C_{\sigma} C Φ = A C σ (321) , it predicts
C σ = 0. (328) C_{\sigma}=0. \tag*{(328)} C σ = 0. (328) Second, the forbidden-side Painlevé action (S274) fixes the sharp RBM tail rate:
− log σ 2 ( ξ ) ∼ C R B M ∣ ξ ∣ 5 / 4 , C R B M = A − c 5 / 4 = 8 3 5 = 2.771281292 … , ξ → − ∞ . (329) -\log\sigma_{2}(\xi)\sim C_{\mathrm{RBM}}|\xi|^{5/4},\qquad C_{\mathrm{RBM}}=A_{-c}^{5/4}=\frac{8\sqrt{3}}{5}=2.771281292\ldots,\qquad\xi\to-\infty. \tag*{(329)} − log σ 2 ( ξ ) ∼ C RBM ∣ ξ ∣ 5/4 , C RBM = A − c 5/4 = 5 8 3 = 2.771281292 … , ξ → − ∞. (329) The vertical metric A A A O ( 1 ) O(1) O ( 1 ) − log σ 2 -\log\sigma_{2} − log σ 2 ∣ ξ ∣ 5 / 4 |\xi|^{5/4} ∣ ξ ∣ 5/4 (329) .
The logical status of the comparison is therefore precise. On the Riccati side, Φ \Phi Φ b Φ = 1 / 288 b_{\Phi}=1/288 b Φ = 1/288 4 / 5 4/5 4/5 5 / 4 5/4 5/4 b R B M = 1 / 72 b_{\mathrm{RBM}}=1/72 b RBM = 1/72 3 / 2 3/2 3/2 ξ − 7 / 2 \xi^{-7/2} ξ − 7/2
Details of the random-band diagram calculations
For completeness, we give here the diagram calculations entering Eqs. (320) and (325) . Besides providing the numerical coefficients, this makes explicit the normalization conventions of the Liu–Zou expansion and, in particular, the factor of two associated with the two spectral edges.
From averaged polynomial traces to ribbon graphs.— As described in § 0.16 C, Liu and Zou probe the edge through disorder averages of normalized traces of high-degree Chebyshev-type polynomials, which in the complex Gaussian case reduce to ⟨ n − 1 Tr U n ( H / 2 ) ⟩ \langle n^{-1}\operatorname{Tr}U_n(H/2)\rangle ⟨ n − 1 Tr U n ( H /2 )⟩ H H H w e w_e w e Γ \Gamma Γ
The resulting construction may be summarized schematically as
averaged polynomial trace ⟶ \longrightarrow ⟶ ⟶ \longrightarrow ⟶ ⟶ \longrightarrow ⟶ ⟶ \longrightarrow ⟶
The cyclic ordering inherited from the trace supplies the ribbon structure. If V V V E E E F F F g g g ∣ V ∣ − ∣ E ∣ + ∣ F ∣ = 2 − 2 g |V|-|E|+|F|=2-2g ∣ V ∣ − ∣ E ∣ + ∣ F ∣ = 2 − 2 g ϕ 3 \phi^{3} ϕ 3 [67 ] .
Diffusive graph weights and the theta graph.— The variance profile S x y = ⟨ ∣ H x y ∣ 2 ⟩ S_{xy}=\langle|H_{xy}|^{2}\rangle S x y = ⟨ ∣ H x y ∣ 2 ⟩
p α ( x ) = 1 2 π ζ 2 α exp [ − x 2 2 ζ 2 α ] , (330) p_{\alpha}(x)=\frac{1}{\sqrt{2\pi\zeta^{2}\alpha}}\exp\left[-\frac{x^{2}}{2\zeta^{2}\alpha}\right], \tag*{(330)} p α ( x ) = 2 π ζ 2 α 1 exp [ − 2 ζ 2 α x 2 ] , (330) where α \alpha α ζ 2 \zeta^{2} ζ 2 Γ \Gamma Γ
U Γ ( α ) = ∑ T spanning tree ∏ e ∉ T α e , (331) \mathcal{U}_{\Gamma}(\alpha)=\sum_{\substack{T\ \text{spanning tree}}}\prod_{e\notin T}\alpha_{e}, \tag*{(331)} U Γ ( α ) = T spanning tree ∑ e ∈ / T ∏ α e , (331) which is Eq. (3.5) of Ref. [67 ] . In one spatial dimension the Gaussian integrations over internal vertices produce U Γ − 1 / 2 \mathcal{U}_{\Gamma}^{-1/2} U Γ − 1/2
It is useful to see this explicitly for the theta graph: Fixing the translational zero mode by placing A A A x x x B B B
Figure 1 (PDF p. 55).
I θ = ∫ R d x p α 1 ( x ) p α 2 ( x ) p α 3 ( x ) = 1 2 π ς 2 α 1 α 2 + α 1 α 3 + α 2 α 3 , (332) I_{\theta}=\int_{\mathbb{R}}\mathrm{d}x\,p_{\alpha_1}(x)p_{\alpha_2}(x)p_{\alpha_3}(x)
=\frac{1}{2\pi\varsigma^2\sqrt{\alpha_1\alpha_2+\alpha_1\alpha_3+\alpha_2\alpha_3}}, \tag*{(332)} I θ = ∫ R d x p α 1 ( x ) p α 2 ( x ) p α 3 ( x ) = 2 π ς 2 α 1 α 2 + α 1 α 3 + α 2 α 3 1 , (332) in agreement with U θ = α 1 α 2 + α 1 α 3 + α 2 α 3 U_{\theta}=\alpha_1\alpha_2+\alpha_1\alpha_3+\alpha_2\alpha_3 U θ = α 1 α 2 + α 1 α 3 + α 2 α 3
The graph needed for the first shape-sensitive coefficient is obtained by subdividing one theta branch:
Figure 2.
If y y y C C C ∫ d y p α 1 ( y ) p α 2 ( x − y ) = p α 1 + α 2 ( x ) \int\mathrm{d}y\,p_{\alpha_1}(y)p_{\alpha_2}(x-y)=p_{\alpha_1+\alpha_2}(x) ∫ d y p α 1 ( y ) p α 2 ( x − y ) = p α 1 + α 2 ( x )
I s u b = 1 2 π ς 2 α 3 α 4 + ( α 1 + α 2 ) ( α 3 + α 4 ) , (333) I_{\mathrm{sub}}=\frac{1}{2\pi\varsigma^2\sqrt{\alpha_3\alpha_4+(\alpha_1+\alpha_2)(\alpha_3+\alpha_4)}}, \tag*{(333)} I sub = 2 π ς 2 α 3 α 4 + ( α 1 + α 2 ) ( α 3 + α 4 ) 1 , (333) so that U s u b = α 3 α 4 + ( α 1 + α 2 ) ( α 3 + α 4 ) U_{\mathrm{sub}}=\alpha_3\alpha_4+(\alpha_1+\alpha_2)(\alpha_3+\alpha_4) U sub = α 3 α 4 + ( α 1 + α 2 ) ( α 3 + α 4 )
Marked tail and the coefficient b R B M b_{\mathrm{RBM}} b RBM — In the Liu–Zou one-point convention the subdivided-theta core carries one additional marked tail:
Figure S4. First shape-sensitive one-point ribbon diagram in the Liu–Zou convention. Removing the marked tail w 0 w_0 w 0
The full graph has four vertices, five edges and one face, hence genus g = 1 g=1 g = 1 n n n
2 ( w 0 + w 1 + w 2 + w 3 + w 4 ) = n , w 0 = n 2 − ∑ i = 1 4 w i . (334) 2(w_0+w_1+w_2+w_3+w_4)=n,\qquad w_0=\frac{n}{2}-\sum_{i=1}^{4}w_i. \tag*{(334)} 2 ( w 0 + w 1 + w 2 + w 3 + w 4 ) = n , w 0 = 2 n − i = 1 ∑ 4 w i . (334) Eliminating w 0 w_0 w 0 ∑ i = 1 4 w i ≤ n / 2 \sum_{i=1}^{4}w_i\le n/2 ∑ i = 1 4 w i ≤ n /2 D ⊂ R 5 D\subset\mathbb{R}^{5} D ⊂ R 5 (334) . Its induced Euclidean measure is d V o l D = 5 d w 1 ⋯ d w 4 \mathrm{dVol}_D=\sqrt{5}\,\mathrm{d}w_1\cdots\mathrm{d}w_4 dVol D = 5 d w 1 ⋯ d w 4 D D D C D = 1 / 5 C_D=1/\sqrt{5} C D = 1/ 5
C D d V o l D = d w 1 d w 2 d w 3 d w 4 . (335) C_D\,\mathrm{dVol}_D=\mathrm{d}w_1\,\mathrm{d}w_2\,\mathrm{d}w_3\,\mathrm{d}w_4. \tag*{(335)} C D dVol D = d w 1 d w 2 d w 3 d w 4 . (335) Thus the hyperplane factor cancels exactly: the marked tail introduces no additional graph multiplicity.
Passing to continuum lengths and using Eq. (333) , the core contribution before the explicit 1 / n 1/n 1/ n
I Γ ( τ ) = 1 2 π ς 2 ∫ α e ≥ 0 ∑ e = 1 2 α e ≤ τ d 4 α α 3 α 4 + ( α 1 + α 2 ) ( α 3 + α 4 ) . (336) I_{\Gamma}(\tau)=\frac{1}{2\pi\varsigma^2}\int_{\substack{\alpha_e\ge0\\ \sum_{e=1}^{2}\alpha_e\le\tau}}\frac{\mathrm{d}^4\alpha}{\sqrt{\alpha_3\alpha_4+(\alpha_1+\alpha_2)(\alpha_3+\alpha_4)}}. \tag*{(336)} I Γ ( τ ) = 2 π ς 2 1 ∫ α e ≥ 0 ∑ e = 1 2 α e ≤ τ α 3 α 4 + ( α 1 + α 2 ) ( α 3 + α 4 ) d 4 α . (336) With α e = ( τ / 2 ) u e \alpha_e=(\tau/2)u_e α e = ( τ /2 ) u e
I Γ ( τ ) = τ 3 16 π ς 2 J , J = ∫ u i ≥ 0 ∑ i u i ≤ 1 d 4 u u 3 u 4 + ( u 1 + u 2 ) ( u 3 + u 4 ) . (337) I_{\Gamma}(\tau)=\frac{\tau^3}{16\pi\varsigma^2}J,\qquad J=\int_{\substack{u_i\ge0\\ \sum_i u_i\le1}}\frac{\mathrm{d}^4u}{\sqrt{u_3u_4+(u_1+u_2)(u_3+u_4)}}. \tag*{(337)} I Γ ( τ ) = 16 π ς 2 τ 3 J , J = ∫ u i ≥ 0 ∑ i u i ≤ 1 u 3 u 4 + ( u 1 + u 2 ) ( u 3 + u 4 ) d 4 u . (337) The factor 1 / n 1/n 1/ n τ \tau τ τ 2 J / ( 16 π ς 2 ) \tau^2J/(16\pi\varsigma^2) τ 2 J / ( 16 π ς 2 ) s = u 1 + u 2 s=u_1+u_2 s = u 1 + u 2 r = u 3 + u 4 r=u_3+u_4 r = u 3 + u 4 u 3 = r v u_3=rv u 3 = r v u 4 = r ( 1 − v ) u_4=r(1-v) u 4 = r ( 1 − v ) s s s v v v
J = 2 3 ∫ 0 1 ( 1 − z ) arcsin z 4 − 3 z d z = π 27 , (338) \begin{aligned}
J&=\frac{2}{3}\int_0^1(1-z)\arcsin\sqrt{\frac{z}{4-3z}}\,\mathrm{d}z\\
&=\frac{\pi}{27}, \tag*{(338)}
\end{aligned} J = 3 2 ∫ 0 1 ( 1 − z ) arcsin 4 − 3 z z d z = 27 π , (338) where the last equality follows by integration by parts and z = sin 2 θ z=\sin^2\theta z = sin 2 θ
F Γ ( τ ) = τ 2 432 ς 2 + o ( τ 2 ) . (339) F_{\Gamma}(\tau)=\frac{\tau^2}{432\varsigma^2}+o(\tau^2). \tag*{(339)} F Γ ( τ ) = 432 ς 2 τ 2 + o ( τ 2 ) . (339) For Sodin’s sharp-cutoff profile the rescaled longitudinal step is uniform on [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] ς 2 = 1 2 ∫ − 1 1 x 2 d x = 1 / 3 \varsigma^2=\frac{1}{2}\int_{-1}^{1}x^2\,\mathrm{d}x=1/3 ς 2 = 2 1 ∫ − 1 1 x 2 d x = 1/3
F Γ ( τ ) = τ 2 144 + o ( τ 2 ) . (340) F_{\Gamma}(\tau)=\frac{\tau^2}{144}+o(\tau^2). \tag*{(340)} F Γ ( τ ) = 144 τ 2 + o ( τ 2 ) . (340) At this order F Γ F_{\Gamma} F Γ τ 2 \tau^2 τ 2 ϕ 1 r e g \phi_1^{\mathrm{reg}} ϕ 1 reg (317) , with its explicit prefactor 2. After subtracting the leading edge background, integration by parts and sin ( τ ξ ) / ( τ ξ ) = 1 − τ 2 ξ / 6 + O ( τ 4 ξ 2 ) \sin(\tau\sqrt{\xi})/(\tau\sqrt{\xi})=1-\tau^2\xi/6+O(\tau^4\xi^2) sin ( τ ξ ) / ( τ ξ ) = 1 − τ 2 ξ /6 + O ( τ 4 ξ 2 )
[ τ 2 coefficient of ϕ 1 r e g ] = 1 3 ∫ − ∞ ∞ δ σ 2 ( ξ ) d ξ = 1 2 b R B M . (341) \left[\tau^2\text{ coefficient of }\phi_1^{\mathrm{reg}}\right]=\frac{1}{3}\int_{-\infty}^{\infty}\delta\sigma_2(\xi)\,\mathrm{d}\xi=\frac{1}{2}b_{\mathrm{RBM}}. \tag*{(341)} [ τ 2 coefficient of ϕ 1 reg ] = 3 1 ∫ − ∞ ∞ δ σ 2 ( ξ ) d ξ = 2 1 b RBM . (341) Equation (S306) therefore yields
b R B M = 1 72 , (342) b_{\mathrm{RBM}}=\frac{1}{72}, \tag*{(342)} b RBM = 72 1 , (342) which is the result used in Eq. (320) . Thus J = π / 27 J=\pi/27 J = π /27 b R B M = 1 / 72 b_{\mathrm{RBM}}=1/72 b RBM = 1/72
Two independent normalization checks.— Because the metric factors depend directly on b R B M b_{\mathrm{RBM}} b RBM
c 1 = 0 , c 2 = 1 384 . (343) c_1=0,\qquad c_2=\frac{1}{384}. \tag*{(343)} c 1 = 0 , c 2 = 384 1 . (343) This tests the graph multiplicity, the double traversal of every edge and the two-edge convention. Second, direct measurements of the Liu–Zou transform for sharp-cutoff complex-Hermitian band matrices with W = 16 W=16 W = 16 48 48 48
[ τ 2 coefficient of ϕ 1 r e g ] n u m = 0.0065 ( 8 ) , 1 144 = 0.006944 … , (344) \left[\tau^2\text{ coefficient of }\phi_1^{\mathrm{reg}}\right]_{\mathrm{num}}=0.0065(8),\qquad\frac{1}{144}=0.006944\ldots, \tag*{(344)} [ τ 2 coefficient of ϕ 1 reg ] num = 0.0065 ( 8 ) , 144 1 = 0.006944 … , (344) consistent with Eq. (340) .
The first nonanalytic coefficient
The preceding genus-one calculation supplies the regular coefficient used to fix the metric. A logically independent test comes from the first nonanalytic term of the unitary ribbon expansion. Write this term in the regularized one-point transform as ϕ 1 r e g ( τ ) ⊃ C 4 ∣ τ ∣ 7 \phi_1^{\mathrm{reg}}(\tau) \supset C_4|\tau|^7 ϕ 1 reg ( τ ) ⊃ C 4 ∣ τ ∣ 7 ∣ τ ∣ 7 , ∣ τ ∣ 17 , ∣ τ ∣ 27 , … |\tau|^7,|\tau|^{17},|\tau|^{27},\ldots ∣ τ ∣ 7 , ∣ τ ∣ 17 , ∣ τ ∣ 27 , … ξ − 7 / 2 , ξ − 17 / 2 , ξ − 27 / 2 , … \xi^{-7/2},\xi^{-17/2},\xi^{-27/2},\ldots ξ − 7/2 , ξ − 17/2 , ξ − 27/2 , …
For the first such contribution the reduced one-point diagram has
V = 8 , E = 11 , F = 1 , 2 − 2 g = V − E + F = − 2 , (345) V=8,\qquad E=11,\qquad F=1,\qquad2-2g=V-E+F=-2, \tag*{(345)} V = 8 , E = 11 , F = 1 , 2 − 2 g = V − E + F = − 2 , (345) and hence g = 2 g=2 g = 2
For k ≥ 1 k\ge1 k ≥ 1 Δ k = { ( x 1 , … , x k + 1 ) ∈ R + k + 1 : ∑ e = 1 k + 1 x e = 1 } \Delta_k=\{(x_1,\ldots,x_{k+1})\in\mathbb{R}_+^{k+1}:\sum_{e=1}^{k+1}x_e=1\} Δ k = {( x 1 , … , x k + 1 ) ∈ R + k + 1 : ∑ e = 1 k + 1 x e = 1 } k k k k k k D \mathcal{D} D
J D = ∫ Δ 10 d 10 x U D ( x ) . (346) J_{\mathcal{D}}=\int_{\Delta_{10}}\frac{\mathrm{d}^{10}x}{\sqrt{U_{\mathcal{D}}(x)}}. \tag*{(346)} J D = ∫ Δ 10 U D ( x ) d 10 x . (346) The root edge is a bridge and may be integrated explicitly. After suppressing the resulting series pair one is left with a six-vertex, nine-edge cubic multigraph G G G M 2 ( 3 ) \mathcal{M}^{(3)}_2 M 2 ( 3 ) G ( M ) G(\mathcal{M}) G ( M ) M ∈ M 2 ( 3 ) \mathcal{M}\in\mathcal{M}^{(3)}_2 M ∈ M 2 ( 3 )
I G = ∫ Δ 8 U G ( α ) − 1 / 2 d 8 α , ∑ D J D = 1 72 ∑ M ∈ M 2 ( 3 ) I G ( M ) . (347) I_G=\int_{\Delta_8}U_G(\alpha)^{-1/2}\,\mathrm{d}^8\alpha,\qquad\sum_{\mathcal{D}}J_{\mathcal{D}}=\frac{1}{72}\sum_{\mathcal{M}\in\mathcal{M}^{(3)}_2}I_{G(\mathcal{M})}. \tag*{(347)} I G = ∫ Δ 8 U G ( α ) − 1/2 d 8 α , D ∑ J D = 72 1 M ∈ M 2 ( 3 ) ∑ I G ( M ) . (347) These one-dimensional graph integrals can be evaluated particularly simply in position space. Introduce four independent loop momenta p 1 , … , p 4 p_1,\ldots,p_4 p 1 , … , p 4 q e q_e q e p ℓ p_\ell p ℓ
I G = ∫ ∏ ℓ = 1 4 d p ℓ 2 π ∏ e = 1 9 1 q e 2 + 1 . (348) I_G=\int\prod_{\ell=1}^{4}\frac{\mathrm{d}p_\ell}{2\pi}\prod_{e=1}^{9}\frac{1}{q_e^2+1}. \tag*{(348)} I G = ∫ ℓ = 1 ∏ 4 2 π d p ℓ e = 1 ∏ 9 q e 2 + 1 1 . (348) Schwinger parametrization gives I G = ( 45 / π 2 ) I G I_G=(45/\pi^2)I_G I G = ( 45/ π 2 ) I G ∫ d p e i p x / [ 2 π ( p 2 + 1 ) ] = e − ∣ x ∣ / 2 \int\mathrm{d}p\,e^{ipx}/[2\pi(p^2+1)]=e^{-|x|}/2 ∫ d p e i p x / [ 2 π ( p 2 + 1 )] = e − ∣ x ∣ /2 x 6 = 0 x_6=0 x 6 = 0
I G = π 2 23040 K G , K G = ∫ R 5 exp [ − ∑ ( u v ) ∈ E ( G ) ∣ x u − x v ∣ ] d x 1 ⋯ d x 5 . (349) I_G=\frac{\pi^2}{23040}K_G,\qquad K_G=\int_{\mathbb{R}^5}\exp\left[-\sum_{(uv)\in E(G)}|x_u-x_v|\right]\mathrm{d}x_1\cdots\mathrm{d}x_5. \tag*{(349)} I G = 23040 π 2 K G , K G = ∫ R 5 exp − ( uv ) ∈ E ( G ) ∑ ∣ x u − x v ∣ d x 1 ⋯ d x 5 . (349) Ordering the six vertex coordinates, including the fixed coordinate x 6 = 0 x_6=0 x 6 = 0 c j ( π ) c_j(\pi) c j ( π ) j j j 6 − j 6-j 6 − j π ∈ S 6 \pi\in S_6 π ∈ S 6
K G = ∑ π ∈ S 6 ∏ j = 1 5 1 c j ( π ) . (350) K_G=\sum_{\pi\in S_6}\prod_{j=1}^{5}\frac{1}{c_j(\pi)}. \tag*{(350)} K G = π ∈ S 6 ∑ j = 1 ∏ 5 c j ( π ) 1 . (350) Thus every graph integral reduces to a finite rational sum. To make the five graph classes explicit, choose vertices 1 , … , 6 1,\ldots,6 1 , … , 6
E 1 = { 15 , 16 , 16 , 23 , 24 , 25 , 34 , 34 , 56 } , E 2 = { 15 , 16 , 16 , 24 , 25 , 26 , 34 , 34 , 35 } , E 3 = { 15 , 16 , 16 , 23 , 24 , 26 , 34 , 35 , 45 } , E 4 = { 14 , 15 , 16 , 23 , 25 , 26 , 34 , 36 , 45 } , E 5 = { 14 , 15 , 16 , 24 , 25 , 26 , 34 , 35 , 36 } . (351) \begin{aligned}
E_1&=\{15,16,16,23,24,25,34,34,56\},\\
E_2&=\{15,16,16,24,25,26,34,34,35\},\\
E_3&=\{15,16,16,23,24,26,34,35,45\},\\
E_4&=\{14,15,16,23,25,26,34,36,45\},\\
E_5&=\{14,15,16,24,25,26,34,35,36\}. \tag*{(351)}
\end{aligned} E 1 E 2 E 3 E 4 E 5 = { 15 , 16 , 16 , 23 , 24 , 25 , 34 , 34 , 56 } , = { 15 , 16 , 16 , 24 , 25 , 26 , 34 , 34 , 35 } , = { 15 , 16 , 16 , 23 , 24 , 26 , 34 , 35 , 45 } , = { 14 , 15 , 16 , 23 , 25 , 26 , 34 , 36 , 45 } , = { 14 , 15 , 16 , 24 , 25 , 26 , 34 , 35 , 36 } . (351) Here, for example, 16 denotes an edge joining vertices 1 and 6. For these five classes one obtains
Table 1.
∑ M ∈ M 2 ( 3 ) K G ( M ) = 896 9 . (352) \sum_{\mathcal{M}\in\mathfrak{M}^{(3)}_2} K_G(\mathcal{M})=\frac{896}{9}. \tag*{(352)} M ∈ M 2 ( 3 ) ∑ K G ( M ) = 9 896 . (352) Combining the graph multiplicities, the continuum normalization and Eq. (349) gives the coefficient C 4 C_4 C 4 ∣ τ ∣ 7 |\tau|^7 ∣ τ ∣ 7
C 4 = 9 1024 π 2 1 72 π 2 23040 896 9 = 7 13271040 . (353) C_4=\frac{9}{1024\pi^2}\frac{1}{72}\frac{\pi^2}{23040}\frac{896}{9}=\frac{7}{13271040}. \tag*{(353)} C 4 = 1024 π 2 9 72 1 23040 π 2 9 896 = 13271040 7 . (353) To convert this two-edge transform coefficient to the one-edge cumulative intensity, suppose δ σ 2 ( ξ ) ⊃ a R B M ξ − 7 / 2 \delta\sigma_2(\xi)\supset a_{\mathrm{RBM}}\xi^{-7/2} δ σ 2 ( ξ ) ⊃ a RBM ξ − 7/2 (317) then gives
C 4 = − π 5760 a R B M , (354) C_4=-\frac{\pi}{5760}a_{\mathrm{RBM}}, \tag*{(354)} C 4 = − 5760 π a RBM , (354) where the denominator 5760 includes the factor 2 from the two equivalent spectral edges. Hence
a R B M = − 7 2304 π . (355) a_{\mathrm{RBM}}=-\frac{7}{2304\pi}. \tag*{(355)} a RBM = − 2304 π 7 . (355) This is the coefficient quoted in Eq. (325) . Since the metric factors c c c A \mathcal{A} A 3 / 2 3/2 3/2 b Φ / b R B M b_{\Phi}/b_{\mathrm{RBM}} b Φ / b RBM (355) with the Painlevé-I X − 7 / 2 X^{-7/2} X − 7/2
Finite-length propagator and the lowest-eigenvalue distribution
§ 0.5 and II already identify the finite-length Dirichlet level count with the total projective winding count of the Riccati eigenvalues, and describe the cyclic reinjection at λ = ± ∞ \lambda=\pm\infty λ = ± ∞
The central object is a transition density in which every projective passage is assigned a factor z z z z = 1 z=1 z = 1 z = e i χ z=e^{i\chi} z = e i χ z = 0 z=0 z = 0 E E E β = 2 \beta=2 β = 2 z z z z = 0 z=0 z = 0 N N N
Counting-weighted propagator
We first construct the finite-length object from which both the spectral counting statistics and the lowest-eigenvalue distribution follow. Fix E E E μ = ( μ 1 , … , μ N ) \boldsymbol{\mu}=(\mu_1,\ldots,\mu_N) μ = ( μ 1 , … , μ N ) C N C_N C N K x ( E ) K_x(E) K x ( E ) ( 0 , x ] (0,x] ( 0 , x ] § 0.5 .
For a complex counting parameter z z z
P x [ z ] ( λ ∣ μ ) d N λ = E μ [ z K x ( E ) 1 { λ ( x ) ∈ d N λ } ] . (356) P_x^{[z]}(\boldsymbol{\lambda}\mid\boldsymbol{\mu})\,\mathrm{d}^N\lambda
=\mathbb{E}_{\boldsymbol{\mu}}\left[z^{K_x(E)}\mathbf{1}_{\{\boldsymbol{\lambda}(x)\in\mathrm{d}^N\lambda\}}\right]. \tag*{(356)} P x [ z ] ( λ ∣ μ ) d N λ = E μ [ z K x ( E ) 1 { λ ( x ) ∈ d N λ } ] . (356) Here E μ \mathbb{E}_{\boldsymbol{\mu}} E μ λ ( 0 ) = μ \boldsymbol{\lambda}(0)=\boldsymbol{\mu} λ ( 0 ) = μ z = 1 z=1 z = 1 z = e i χ z=e^{i\chi} z = e i χ e i χ e^{i\chi} e i χ z = 0 z=0 z = 0
The bulk dynamics is unchanged by the factor z K x z^{K_x} z K x K x K_x K x β = 2 \beta= 2 β = 2
V N ( λ ) = ∏ i < j ( λ j − λ i ) , C N = { λ 1 < ⋯ < λ N } , (357) V_N(\lambda) = \prod_{i<j}(\lambda_j-\lambda_i), \qquad C_N = \{\lambda_1 < \cdots< \lambda_N\}, \tag*{(357)} V N ( λ ) = i < j ∏ ( λ j − λ i ) , C N = { λ 1 < ⋯ < λ N } , (357) the transformation P = V N Ψ P = V_N\Psi P = V N Ψ N N N
ℓ N = T ∂ λ 2 + ( E + λ 2 ) ∂ λ + ( N + 1 ) λ , (358) \ell_N = T\partial_\lambda^2 + (E+\lambda^2)\partial_\lambda+ (N+1)\lambda, \tag*{(358)} ℓ N = T ∂ λ 2 + ( E + λ 2 ) ∂ λ + ( N + 1 ) λ , (358) which is the operator already obtained in Eq. (68) .
Let G x [ z ] ( λ , μ ) G_x^{[z]}(\lambda,\mu) G x [ z ] ( λ , μ )
∂ x G x [ z ] ( λ , μ ) = ℓ N ( λ ) G x [ z ] ( λ , μ ) , G 0 [ z ] ( λ , μ ) = δ ( λ − μ ) . (359) \partial_xG_x^{[z]}(\lambda,\mu)=\ell_N^{(\lambda)}G_x^{[z]}(\lambda,\mu), \qquad G_0^{[z]}(\lambda,\mu)=\delta(\lambda-\mu). \tag*{(359)} ∂ x G x [ z ] ( λ , μ ) = ℓ N ( λ ) G x [ z ] ( λ , μ ) , G 0 [ z ] ( λ , μ ) = δ ( λ − μ ) . (359) Then the weighted N N N
P x [ z ] ( λ ∣ μ ) = V N ( λ ) V N ( μ ) det 1 ≤ i , j ≤ N [ G x [ z ] ( λ i , μ j ) ] . (360) P_x^{[z]}(\lambda\mid\mu)=\frac{V_N(\lambda)}{V_N(\mu)}\det_{1\le i,j\le N}\left[G_x^{[z]}(\lambda_i,\mu_j)\right]. \tag*{(360)} P x [ z ] ( λ ∣ μ ) = V N ( μ ) V N ( λ ) 1 ≤ i , j ≤ N det [ G x [ z ] ( λ i , μ j ) ] . (360) The determinant solves the fermionized bulk equation, vanishes when two Riccati eigenvalues collide, and together with the Vandermonde ratio reproduces the delta-function initial condition in the ordered chamber. What remains is to specify how the counting factor z z z
Counting weight at the projective boundary. — The global geometry needed here has already been established in Sec. II. Recall only that λ = cot θ \lambda=\cot\theta λ = cot θ θ = 0 \theta=0 θ = 0 θ = π \theta=\pi θ = π R P 1 \mathbb{RP}^{1} RP 1 (57) . The corresponding compactified one-body orbitals carry the fermionic factor
σ N = ( − 1 ) N − 1 (361) \sigma_N=(-1)^{N-1} \tag*{(361)} σ N = ( − 1 ) N − 1 (361) under one circuit of the projective circle, as shown in Eq. (61) .
To state the same condition for the propagator, write λ = cot θ \lambda=\cot\theta λ = cot θ μ = cot φ \mu=\cot\varphi μ = cot φ
G ^ x [ z ] ( θ , φ ) = sin − ( N + 1 ) θ G x [ z ] ( cot θ , cot φ ) sin N − 1 φ . (362) \widehat{G}_x^{[z]}(\theta,\varphi)=\sin^{-(N+1)}\theta\,G_x^{[z]}(\cot\theta,\cot\varphi)\sin^{N-1}\varphi. \tag*{(362)} G x [ z ] ( θ , φ ) = sin − ( N + 1 ) θ G x [ z ] ( cot θ , cot φ ) sin N − 1 φ . (362) For fixed initial point φ \varphi φ z z z
G ^ x [ z ] ( 0 , φ ) = σ N z G ^ x [ z ] ( π , φ ) , σ N = ( − 1 ) N − 1 . (363) \widehat{G}_x^{[z]}(0,\varphi)=\sigma_N z\,\widehat{G}_x^{[z]}(\pi,\varphi), \qquad\sigma_N=(-1)^{N-1}. \tag*{(363)} G x [ z ] ( 0 , φ ) = σ N z G x [ z ] ( π , φ ) , σ N = ( − 1 ) N − 1 . (363) No modification of the differential operator Eq. (359) is involved: the counting field enters entirely through this global boundary condition.
It remains to check that Eq. (363) indeed assigns the factor z z z (360) . This produces a second factor ( − 1 ) N − 1 = σ N (-1)^{N-1}=\sigma_N ( − 1 ) N − 1 = σ N (363) therefore cancels the permutation sign, while the circular Vandermonde is continuous under the same cyclic gluing, as shown in Eq. (59) . The complete many-body propagator consequently acquires exactly
σ N 2 z = z (364) \sigma_N^2z=z \tag*{(364)} σ N 2 z = z (364) at each passage. After K x ( E ) K_x(E) K x ( E ) z K x ( E ) z^{K_x(E)} z K x ( E ) (356) of Eq. (360) .
The three boundary realizations needed below are consequently transparent. At z = 1 z=1 z = 1 z = e i χ z=e^{i\chi} z = e i χ
G x χ ( λ , μ ) : = G x [ e i χ ] ( λ , μ ) . (365) G_x^\chi(\lambda,\mu):=G_x^{[e^{i\chi}]}(\lambda,\mu). \tag*{(365)} G x χ ( λ , μ ) := G x [ e i χ ] ( λ , μ ) . (365) Finally, at z = 0 z=0 z = 0
Dirichlet entrance and the finite-length level count
For the spectral problem the fundamental matrix obeys Y ( 0 ) = 0 Y(0)=0 Y ( 0 ) = 0 Y ′ ( 0 ) = I N Y'(0)=I_N Y ′ ( 0 ) = I N Z ( x ) = Y ′ ( x ) Y − 1 ( x ) ≃ I N / x Z(x)=Y'(x)Y^{-1}(x)\simeq I_N/x Z ( x ) = Y ′ ( x ) Y − 1 ( x ) ≃ I N / x x ↓ 0 x\downarrow0 x ↓ 0 N N N + ∞ +\infty + ∞ (360) is written for distinct points in the ordered chamber, this degenerate entrance must be understood as a confluent limit.
More precisely, for fixed M M M
μ i = M + ϵ ξ i , ξ 1 < ⋯ < ξ N , (366) \mu_i=M+\epsilon\xi_i,\qquad\xi_1<\cdots<\xi_N, \tag*{(366)} μ i = M + ϵ ξ i , ξ 1 < ⋯ < ξ N , (366) so that μ ∈ C N \boldsymbol{\mu}\in\mathcal{C}_N μ ∈ C N ϵ > 0 \epsilon>0 ϵ > 0 ϵ → 0 + \epsilon\to0^+ ϵ → 0 + μ 1 , … , μ N → M \mu_1,\ldots,\mu_N\to M μ 1 , … , μ N → M Z ( 0 ) = M I N Z(0)=MI_N Z ( 0 ) = M I N Y ( 0 ) = I N Y(0)=I_N Y ( 0 ) = I N Y ′ ( 0 ) = M I N Y'(0)=MI_N Y ′ ( 0 ) = M I N Y ↦ M − 1 Y Y\mapsto M^{-1}Y Y ↦ M − 1 Y M → + ∞ M\to+\infty M → + ∞ Y ( 0 ) = 0 Y(0)=0 Y ( 0 ) = 0 Y ′ ( 0 ) = I N Y'(0)=I_N Y ′ ( 0 ) = I N lim μ → + ∞ D \lim_{\boldsymbol{\mu}\to+\infty}^{\mathrm{D}} lim μ → + ∞ D
With the convention that the initial point itself is not counted, matrix oscillation, Sec. I, gives
K L ( E ) = N L ( E ) , (367) K_L(E)=N_L(E), \tag*{(367)} K L ( E ) = N L ( E ) , (367) where N L ( E ) N_L(E) N L ( E ) H L \mathcal{H}_L H L E E E (360) over the final Riccati configuration therefore gives
F N ( z ; E , L ) : = E [ z N L ( E ) ] = lim μ → + ∞ D ∫ C N V N ( λ ) V N ( μ ) det [ G L [ z ] ( λ i , μ j ) ] i , j d N λ . (368) \mathcal{F}_N(z;E,L):=\mathbb{E}\left[z^{N_L(E)}\right]
=\lim_{\boldsymbol{\mu}\to+\infty}^{\mathrm{D}}\int_{\mathcal{C}_N}\frac{V_N(\boldsymbol{\lambda})}{V_N(\boldsymbol{\mu})}\det\left[G_L^{[z]}(\lambda_i,\mu_j)\right]_{i,j}\,\mathrm{d}^N\lambda. \tag*{(368)} F N ( z ; E , L ) := E [ z N L ( E ) ] = μ → + ∞ lim D ∫ C N V N ( μ ) V N ( λ ) det [ G L [ z ] ( λ i , μ j ) ] i , j d N λ . (368) For z = e i χ z=e^{i\chi} z = e i χ (34) of the Main Text. Equivalently,
F N ( z ; E , L ) = ∑ m ≥ 0 z m Pr { N L ( E ) = m } . (369) \mathcal{F}_N(z;E,L)=\sum_{m\geq0}z^m\Pr\{N_L(E)=m\}. \tag*{(369)} F N ( z ; E , L ) = m ≥ 0 ∑ z m Pr { N L ( E ) = m } . (369) Thus the same one-body kernel generates the complete finite-length Dirichlet level-counting distribution. If the Dirichlet eigenvalues are ordered as E 0 ( L ) < E 1 ( L ) < ⋯ E_0(L)<E_1(L)<\cdots E 0 ( L ) < E 1 ( L ) < ⋯
Pr { E k ( L ) ≤ E } = Pr { N L ( E ) ≥ k + 1 } . (370) \Pr\{E_k(L)\leq E\}=\Pr\{N_L(E)\geq k+1\}. \tag*{(370)} Pr { E k ( L ) ≤ E } = Pr { N L ( E ) ≥ k + 1 } . (370) In particular the lowest-eigenvalue distribution is obtained from the zero-count sector.
Zero count and the killed quartic problem
Setting z = 0 z=0 z = 0 (369) selects trajectories with no projective passage:
S N ( E , L ) : = F N ( 0 ; E , L ) = Pr { N L ( E ) = 0 } = Pr { E 0 ( L ) > E } . (371) S_N(E,L):=\mathcal{F}_N(0;E,L)=\Pr\{N_L(E)=0\}=\Pr\{E_0(L)>E\}. \tag*{(371)} S N ( E , L ) := F N ( 0 ; E , L ) = Pr { N L ( E ) = 0 } = Pr { E 0 ( L ) > E } . (371) Thus the survival probability of the Riccati process before its first projective passage is exactly the probability that the finite Schrödinger operator has no eigenvalue below E E E
For z = 0 z=0 z = 0 λ = − ∞ \lambda=-\infty λ = − ∞ λ = + ∞ \lambda=+\infty λ = + ∞ z z z z = 0 z=0 z = 0 λ = − ∞ \lambda=-\infty λ = − ∞ λ = − ∞ \lambda=-\infty λ = − ∞ λ = + ∞ \lambda=+\infty λ = + ∞
The relation to an ordinary Hermitian problem becomes transparent after the standard gauge transformation. With U E ( λ ) = λ 3 / 3 + E λ U_E(\lambda)=\lambda^3/3+E\lambda U E ( λ ) = λ 3 /3 + E λ
e U E / ( 2 T ) L N e − U E / ( 2 T ) = − H N , H N = − T ∂ λ 2 + ( λ 2 + E ) 2 4 T − N λ . (372) e^{U_E/(2T)}\mathcal{L}_Ne^{-U_E/(2T)}=-H_N,\qquad H_N=-T\partial_\lambda^2+\frac{(\lambda^2+E)^2}{4T}-N\lambda. \tag*{(372)} e U E / ( 2 T ) L N e − U E / ( 2 T ) = − H N , H N = − T ∂ λ 2 + 4 T ( λ 2 + E ) 2 − N λ . (372) This identity concerns the bulk differential expression; the boundary realization still has to be specified. For the reinjecting Riccati process the values at λ = − ∞ \lambda=-\infty λ = − ∞ λ = + ∞ \lambda=+\infty λ = + ∞ H N H_N H N
For z = 0 z = 0 z = 0 λ = − ∞ \lambda= -\infty λ = − ∞ λ = + ∞ \lambda= +\infty λ = + ∞ L 2 ( R ) L^2(\mathbb{R}) L 2 ( R )
( λ 2 + E ) 2 4 T − N λ ∼ λ 4 4 T , ∣ λ ∣ → ∞ , (373) \frac{(\lambda^2 + E)^2}{4T} - N\lambda\sim\frac{\lambda^4}{4T}, \qquad|\lambda| \to\infty, \tag*{(373)} 4 T ( λ 2 + E ) 2 − N λ ∼ 4 T λ 4 , ∣ λ ∣ → ∞ , (373) the transformed potential is confining, so H N H_N H N L 2 ( R ) L^2(\mathbb{R}) L 2 ( R )
H N ϕ n = h n ϕ n , h 0 < h 1 < ⋯ , ϕ n ( λ ) ⟶ 0 ( λ → ± ∞ ) , (374) H_N\phi_n = h_n\phi_n,\qquad h_0 < h_1 < \cdots,\qquad\phi_n(\lambda) \longrightarrow0\quad(\lambda\to\pm\infty), \tag*{(374)} H N ϕ n = h n ϕ n , h 0 < h 1 < ⋯ , ϕ n ( λ ) ⟶ 0 ( λ → ± ∞ ) , (374) with the eigenfunctions chosen orthonormal, ∫ R ϕ n ( λ ) ϕ m ( λ ) d λ = δ n m \int_{\mathbb{R}} \phi_n(\lambda)\phi_m(\lambda)\,\mathrm{d}\lambda= \delta_{nm} ∫ R ϕ n ( λ ) ϕ m ( λ ) d λ = δ nm
G x [ 0 ] ( λ , μ ) = e − [ U E ( λ ) − U E ( μ ) ] / ( 2 T ) ∑ n ≥ 0 ϕ n ( λ ) ϕ n ( μ ) e − x h n . (375) G_x^{[0]}(\lambda,\mu) = e^{-[U_E(\lambda)-U_E(\mu)]/(2T)} \sum_{n\geq0}\phi_n(\lambda)\phi_n(\mu)e^{-x h_n}. \tag*{(375)} G x [ 0 ] ( λ , μ ) = e − [ U E ( λ ) − U E ( μ )] / ( 2 T ) n ≥ 0 ∑ ϕ n ( λ ) ϕ n ( μ ) e − x h n . (375) The many-body meaning of this spectral resolution is particularly simple. The killed N N N N N N
H N = ∑ i = 1 N H N ( i ) . (376) \mathbb{H}_N = \sum_{i=1}^{N} H_N^{(i)}. \tag*{(376)} H N = i = 1 ∑ N H N ( i ) . (376) Since the fermions are noninteracting, an N N N N N N
J = { j 1 < ⋯ < j N } , Φ J ( λ ) = det 1 ≤ i , a ≤ N [ ϕ j a ( λ i ) ] , (377) J = \{j_1 < \cdots< j_N\},\qquad\Phi_J(\lambda) = \det_{1\leq i,a\leq N}[\phi_{j_a}(\lambda_i)], \tag*{(377)} J = { j 1 < ⋯ < j N } , Φ J ( λ ) = 1 ≤ i , a ≤ N det [ ϕ j a ( λ i )] , (377) with many-body energy
H J ( E ) = ∑ j ∈ J h j ( E ) . (378) H_J(E) = \sum_{j\in J} h_j(E). \tag*{(378)} H J ( E ) = j ∈ J ∑ h j ( E ) . (378) Thus J J J Φ J \Phi_J Φ J
S N ( E , L ) = ∑ J : ∣ J ∣ = N A J ( E ) e − L H J ( E ) . (379) S_N(E,L) = \sum_{J:\,|J|=N} A_J(E)e^{-L H_J(E)}. \tag*{(379)} S N ( E , L ) = J : ∣ J ∣ = N ∑ A J ( E ) e − L H J ( E ) . (379) The coefficient A J ( E ) A_J(E) A J ( E ) Φ J \Phi_J Φ J
A J = B J C J D , (380) A_J = B_J C_J^{\mathrm{D}}, \tag*{(380)} A J = B J C J D , (380) where, with
m ( λ ) = V N ( λ ) e − ∑ i U E ( λ i ) / ( 2 T ) , (381) m(\lambda) = V_N(\lambda)e^{-\sum_i U_E(\lambda_i)/(2T)}, \tag*{(381)} m ( λ ) = V N ( λ ) e − ∑ i U E ( λ i ) / ( 2 T ) , (381) one has
B J = ∫ C N m ( λ ) Φ J ( λ ) d N λ , C J D = lim μ → + ∞ D Φ J ( μ ) m ( μ ) . (382) B_J = \int_{C_N} m(\lambda)\Phi_J(\lambda)\,\mathrm{d}^N\lambda,\qquad C_J^{\mathrm{D}} = \lim_{\mu\to+\infty}^{\mathrm{D}}\frac{\Phi_J(\mu)}{m(\mu)}. \tag*{(382)} B J = ∫ C N m ( λ ) Φ J ( λ ) d N λ , C J D = μ → + ∞ lim D m ( μ ) Φ J ( μ ) . (382) These coefficients are boundary overlaps, not equilibrium occupation probabilities. The lowest antisymmetric N N N N N N
J 0 = { 0 , 1 , … , N − 1 } . (383) J_0 = \{0,1,\ldots,N-1\}. \tag*{(383)} J 0 = { 0 , 1 , … , N − 1 } . (383) Hence at large L L L
S N ( E , L ) = A 0 ( E ) e − L κ N ( E ) [ 1 + O ( e − L [ h N ( E ) − h N − 1 ( E ) ] ) ] , S_N(E,L)=A_0(E)e^{-L\kappa_N(E)}\left[1+O\left(e^{-L[h_N(E)-h_{N-1}(E)]}\right)\right], S N ( E , L ) = A 0 ( E ) e − L κ N ( E ) [ 1 + O ( e − L [ h N ( E ) − h N − 1 ( E )] ) ] , κ N ( E ) = ∑ n = 0 N − 1 h n ( E ) > 0. (384) \kappa_N(E)=\sum_{n=0}^{N-1}h_n(E)>0. \tag*{(384)} κ N ( E ) = n = 0 ∑ N − 1 h n ( E ) > 0. (384) where A 0 A_0 A 0 J = { 0 , … , N − 1 } J=\{0,\ldots,N-1\} J = { 0 , … , N − 1 } E E E N N N H N H_N H N
It is worth emphasizing the difference between this problem and the stationary one studied in the preceding sections. The differential expression is the same quartic one, but the global boundary condition is different. The stationary problem has projective reinjection and a nonzero winding current; the zero-count problem suppresses reinjection and is therefore an absorbing, or killed, problem. Its decay rate κ N ( E ) \kappa_N(E) κ N ( E ) ν N ( E ) \nu_N(E) ν N ( E )
Collective edge and companion calculation
Equations (368) –(384) are the starting point of the finite-length analysis developed in the companion work [70 ] . There the Dirichlet confluent limit and the boundary overlaps are evaluated explicitly, the finite-N N N
At fixed disorder strength, with T = J 2 / ( 2 N ) T=J^2/(2N) T = J 2 / ( 2 N ) s J = ( 2 J 2 ) 1 / 3 s_J=(2J^2)^{1/3} s J = ( 2 J 2 ) 1/3
E = E ∗ + s J 2 N − 4 / 5 X , E ∗ = − 3 4 s J 2 , ℓ = s J N − 1 / 5 L . (385) E=E^*+s_J^2N^{-4/5}X,\qquad E^*=-\frac{3}{4}s_J^2,\qquad\ell=s_JN^{-1/5}L. \tag*{(385)} E = E ∗ + s J 2 N − 4/5 X , E ∗ = − 4 3 s J 2 , ℓ = s J N − 1/5 L . (385) In this limit the companion calculation constructs a universal survival function S ( ℓ , X ) \mathcal{S}(\ell,X) S ( ℓ , X )
S ( ℓ , X ) ∼ A ( X ) e − ℓ K ( X ) , ℓ → ∞ , (386) \mathcal{S}(\ell,X)\sim A(X)e^{-\ell\mathcal{K}(X)},\qquad\ell\to\infty, \tag*{(386)} S ( ℓ , X ) ∼ A ( X ) e − ℓ K ( X ) , ℓ → ∞ , (386) with the amplitude A ( X ) A(X) A ( X ) K ( X ) \mathcal{K}(X) K ( X ) [70 ] . These quantities contain information beyond the mean density of states studied in the present Letter: the probability of observing zero levels is a full counting observable and is not fixed by the first counting cumulant alone.
Counting statistics, generalized Lyapunov exponents, and joint tilts
The finite-length fermionic formulation of Sec. XII naturally extends to two broader fluctuation problems. A counting tilt probes the full statistics of the number N L ( E ) N_L(E) N L ( E ) E E E ∣ det Y ( L ) ∣ |\det Y(L)| ∣ det Y ( L ) ∣ [27 ] . In both cases the bulk β = 2 \beta=2 β = 2
Spectral counting field
The finite-length counting generating function was constructed in Sec. XII. To characterize its fluctuations as L → ∞ L\to\infty L → ∞ t t t z = e t z=e^t z = e t
Λ N c n t ( t ; E ) : = lim L → ∞ 1 L log E [ e t N L ( E ) ] . (387) \Lambda_N^{\mathrm{cnt}}(t;E):=\lim_{L\to\infty}\frac{1}{L}\log\mathbb{E}\left[e^{tN_L(E)}\right]. \tag*{(387)} Λ N cnt ( t ; E ) := L → ∞ lim L 1 log E [ e t N L ( E ) ] . (387) For large L L L
Let E a ( t ) \mathcal{E}_a(t) E a ( t ) a = 1 , … , N a=1,\ldots,N a = 1 , … , N N N N t = 0 t=0 t = 0
Λ N c n t ( t ; E ) = ∑ a = 1 N ε a ( t ) − ∑ a = 1 N ε a ( 0 ) . (388) \Lambda^{\mathrm{cnt}}_N(t;E)=\sum_{a=1}^{N}\varepsilon_a(t)-\sum_{a=1}^{N}\varepsilon_a(0). \tag*{(388)} Λ N cnt ( t ; E ) = a = 1 ∑ N ε a ( t ) − a = 1 ∑ N ε a ( 0 ) . (388) The subtraction enforces Λ N c n t ( 0 ; E ) = 0 \Lambda^{\mathrm{cnt}}_N(0;E)=0 Λ N cnt ( 0 ; E ) = 0 N N N
Near t = 0 t=0 t = 0 Λ N c n t \Lambda^{\mathrm{cnt}}_N Λ N cnt
∂ t Λ N c n t ( t ; E ) ∣ t = 0 = ν N ( E ) , ∂ t 2 Λ N c n t ( t ; E ) ∣ t = 0 = lim L → ∞ Var N L ( E ) L , (389) \left.\partial_t\Lambda^{\mathrm{cnt}}_N(t;E)\right|_{t=0}=\nu_N(E),\qquad
\left.\partial_t^2\Lambda^{\mathrm{cnt}}_N(t;E)\right|_{t=0}
=\lim_{L\to\infty}\frac{\operatorname{Var}\mathcal{N}_L(E)}{L}, \tag*{(389)} ∂ t Λ N cnt ( t ; E ) t = 0 = ν N ( E ) , ∂ t 2 Λ N cnt ( t ; E ) t = 0 = L → ∞ lim L Var N L ( E ) , (389) provided the dominant branch is differentiable to the indicated order. The present Letter evaluates the first derivative through the stationary winding current; higher counting cumulants are not calculated here.
For the characteristic function one sets locally t = i χ t=i\chi t = i χ Θ N ( χ ; E ) \Theta_N(\chi;E) Θ N ( χ ; E ) χ = 0 \chi=0 χ = 0 χ \chi χ
The zero-count problem of Sec. XII C is another boundary value of the same finite-length generating function. At fixed L L L
lim t → − ∞ E [ e t N L ( E ) ] = S N ( E , L ) = Pr { N L ( E ) = 0 } . (390) \lim_{t\to-\infty}\mathbb{E}\left[e^{t\mathcal{N}_L(E)}\right]
=S_N(E,L)=\Pr\{\mathcal{N}_L(E)=0\}. \tag*{(390)} t → − ∞ lim E [ e t N L ( E ) ] = S N ( E , L ) = Pr { N L ( E ) = 0 } . (390) This identity does not require, and should not be confused with, an interchange of the limits L → ∞ L\to\infty L → ∞ t → − ∞ t\to-\infty t → − ∞ N > 1 N>1 N > 1
Determinant moments and generalized Lyapunov exponents
The second deformation concerns the growth of the Cauchy solution Y ( x ) Y(x) Y ( x ) [27 ] . Between zeros of det Y \det Y det Y
∂ x log ∣ det Y ( x ) ∣ = Tr Z ( x ) . (391) \partial_x\log|\det Y(x)|=\operatorname{Tr}Z(x). \tag*{(391)} ∂ x log ∣ det Y ( x ) ∣ = Tr Z ( x ) . (391) This suggests defining the determinant generalized Lyapunov exponent
Λ N a m p ( q ; E ) : = lim L → ∞ 1 L log E [ ∣ det Y ( L ) ∣ q ] , (392) \Lambda^{\mathrm{amp}}_N(q;E):=\lim_{L\to\infty}\frac{1}{L}\log\mathbb{E}\left[|\det Y(L)|^q\right], \tag*{(392)} Λ N amp ( q ; E ) := L → ∞ lim L 1 log E [ ∣ det Y ( L ) ∣ q ] , (392) whenever the moment and the limit exist.
Formally, the corresponding Feynman–Kac deformation adds q Tr Z q\operatorname{Tr}Z q Tr Z β = 2 \beta=2 β = 2
ℓ N , q † = T ∂ λ 2 − ( E + λ 2 ) ∂ λ + ( N − 1 + q ) λ . (393) \ell^\dagger_{N,q}=T\partial_\lambda^2-(E+\lambda^2)\partial_\lambda+(N-1+q)\lambda. \tag*{(393)} ℓ N , q † = T ∂ λ 2 − ( E + λ 2 ) ∂ λ + ( N − 1 + q ) λ . (393) For N = 1 N=1 N = 1 [27 ] .
For an integer q = m ≥ 0 q=m\geq0 q = m ≥ 0 ℓ N , m † \ell^\dagger_{N,m} ℓ N , m † P N + m − 1 \mathcal{P}_{N+m-1} P N + m − 1 q q q
The difficulty is global rather than local. The identity (S357) holds between zeros of det Y \det Y det Y Y Y Y [27 ] .
For integer m m m
( det Y ( L ) ) m = ∣ det Y ( L ) ∣ m ( − 1 ) m N L ( E ) (394) (\det Y(L))^m = |\det Y(L)|^m(-1)^{m\mathcal{N}_L(E)} \tag*{(394)} ( det Y ( L ) ) m = ∣ det Y ( L ) ∣ m ( − 1 ) m N L ( E ) (394) with the Dirichlet normalization fixing the initial sign. Hence for even m m m m m m ( − 1 ) N L ( E ) (-1)^{\mathcal{N}_L(E)} ( − 1 ) N L ( E ) q q q
Joint spectral and amplitude generator
The counting tilt t t t q q q
Λ N ( q , t ; E ) : = lim L → ∞ 1 L log E [ ∣ det Y ( L ) ∣ q e t N L ( E ) ] . (395) \Lambda_N(q,t;E) := \lim_{L\to\infty} \frac{1}{L}\log\mathbb{E}\left[|\det Y(L)|^q e^{t\mathcal{N}_L(E)}\right]. \tag*{(395)} Λ N ( q , t ; E ) := L → ∞ lim L 1 log E [ ∣ det Y ( L ) ∣ q e t N L ( E ) ] . (395) Where this limit is differentiable at the origin, its first derivatives recover the two observables calculated in the stationary problem,
∂ q Λ N ( q , t ; E ) ∣ q = t = 0 = Γ N ( E ) , ∂ t Λ N ( q , t ; E ) ∣ q = t = 0 = ν N ( E ) , (396) \left.\partial_q\Lambda_N(q,t;E)\right|_{q=t=0}=\Gamma_N(E), \qquad\left.\partial_t\Lambda_N(q,t;E)\right|_{q=t=0}=\nu_N(E), \tag*{(396)} ∂ q Λ N ( q , t ; E ) ∣ q = t = 0 = Γ N ( E ) , ∂ t Λ N ( q , t ; E ) ∣ q = t = 0 = ν N ( E ) , (396) where Γ N ( E ) \Gamma_N(E) Γ N ( E ) ν N ( E ) \nu_N(E) ν N ( E ) log ∣ det Y ( L ) ∣ \log|\det Y(L)| log ∣ det Y ( L ) ∣ N L ( E ) \mathcal{N}_L(E) N L ( E )
∂ q 2 Λ N ∣ 0 = lim L → ∞ Var log ∣ det Y ( L ) ∣ L , \left.\partial_q^2\Lambda_N\right|_0=\lim_{L\to\infty}\frac{\operatorname{Var}\log|\det Y(L)|}{L}, ∂ q 2 Λ N 0 = L → ∞ lim L Var log ∣ det Y ( L ) ∣ , ∂ t 2 Λ N ∣ 0 = lim L → ∞ Var N L ( E ) L , \left.\partial_t^2\Lambda_N\right|_0=\lim_{L\to\infty}\frac{\operatorname{Var}\mathcal{N}_L(E)}{L}, ∂ t 2 Λ N 0 = L → ∞ lim L Var N L ( E ) , ∂ q ∂ t Λ N ∣ 0 = lim L → ∞ Cov ( log ∣ det Y ( L ) ∣ , N L ( E ) ) L , (397) \left.\partial_q\partial_t\Lambda_N\right|_0=\lim_{L\to\infty}\frac{\operatorname{Cov}(\log|\det Y(L)|,\mathcal{N}_L(E))}{L}, \tag*{(397)} ∂ q ∂ t Λ N ∣ 0 = L → ∞ lim L Cov ( log ∣ det Y ( L ) ∣ , N L ( E )) , (397) whenever these limits exist. The counting-only boundary problem has been specified in Sec. XII A, and the zero-count sector q = 0 q=0 q = 0 t → − ∞ t\to-\infty t → − ∞ q q q [70 ] addresses the zero-count sector at q = 0 q=0 q = 0
