Introduction
Determinants are famous for their cancellations; permanents for what happens when those cancellations disappear. For an N × N N \times N N × N M M M
det M = ∑ σ ∈ S N sgn ( σ ) ∏ i M i , σ ( i ) \operatorname{det} M = \sum_{\sigma\in S_N} \operatorname{sgn}(\sigma) \prod_{i} M_{i,\sigma(i)} det M = σ ∈ S N ∑ sgn ( σ ) i ∏ M i , σ ( i ) assigns a sign to each permutation. Swapping two rows or two columns reverses its value, so identical rows or columns make it vanish. This alternating structure permits efficient Gaussian elimination.
In the eighteenth century, Vandermonde studied products of differences that helped shape the theory of alternating functions [1 ] , closely related to the determinant now bearing his name [2 ] . Cauchy subsequently spoke of fonctions symétriques permanentes [3 ] , and it was Muir [4 ] who applied Cauchy’s expression “permanent” to the signless matrix sum
per M = ∑ σ ∈ S N ∏ i = 1 N M i , σ ( i ) . \operatorname{per} M = \sum_{\sigma\in S_N} \prod_{i=1}^{N} M_{i,\sigma(i)}. per M = σ ∈ S N ∑ i = 1 ∏ N M i , σ ( i ) . The missing signs look like a minor edit, yet determinants admit polynomial-time evaluation, while the inclusion-exclusion method uses O ( N 2 N ) O(N 2^N) O ( N 2 N ) [5 , 6 ] . A remarkable difference in complexity.
Valiant proved that exact permanent evaluation is # P \#\mathrm{P} # P [7 ] . The class # P \#\mathrm{P} # P [8 ] .
Caianiello translated the permutation bookkeeping into physical terms. In seminal studies of quantum-field combinatorics, he linked bosonic contractions to signless sums and related pairing structures, placing permanents and Hafnians within the language of quantum particle statistics [9 , 10 , 11 , 12 ] [13 ] . For noninteracting fermions, occupied one-particle orbitals give a Slater determinant [14 ] . The difference between bosonic and fermionic exchange thus gives the old distinction between two permutation sums a concrete physical setting.
Over the last two decades, this connection has become operational in photonics. Scheel showed in 2004 [15 ] , before the boson-sampling proposal of Aaronson and Arkhipov [16 ] , that transition amplitudes of indistinguishable photons in passive linear-optical networks involve permanents of interferometer submatrices. For a given input-output photon pattern, the probability is proportional to the squared modulus of such a permanent. The optical device generates samples without returning the complex permanent: estimating the probability of a very rare output by repeated counting can itself require many trials.
Sampling hardness and evaluating an individual permanent are therefore distinct questions, as is the problem of establishing that a measured distribution agrees with the intended one. The proposed advantage concerns drawing such outputs; its strongest claims for approximate, noisy sampling require additional complexity conjectures [16 ] .
A number of mathematical problems remain open in this context. Tao and Vu focused on the permanents of random Bernoulli matrices, whose entries are independently chosen to be +1 or −1 with equal probability [17 ] . While the determinant moments of matrices with independent Gaussian entries are well understood and can be computed exactly, the Bernoulli case is more intricate, and a systematic study of high-order moments appears to be lacking. In this article, we investigate the moments of permanents for both Gaussian and Bernoulli random matrices.
Integrated photonic experiments made the permanent connection accessible to measurement. Broome and collaborators tested two- and three-photon interference in a tunable six-mode circuit [18 ] ; Spring and collaborators sampled three- and four-photon outputs on a six-mode chip [19 ] . Tillmann and collaborators studied three photons in five-mode written waveguides [20 ] , while Crespi and collaborators realized a laser-written five-mode interferometer and compared three-photon counts with permanent-based predictions [21 ] . These experiments test the underlying interference at small photon numbers; they do not themselves establish a classical-computation speedup.
Validation studies subsequently tested specific alternatives to genuine bosonic sampling [22 ] . In particular, Walschaers et al. proposed efficiently accessible statistics of two-mode photon-number correlations as a benchmark against distinguishable-particle and selected mimic models [23 ] ; related signatures were later measured on an integrated chip [24 ] .
A boson sampling experiment is deceptively simple: a certain number of photons are injected into the input ports, interact with a linear system, and ultimately are detected at the output ports. See the schematic in Figure 1 . However, the underlying problem is extremely complex, involving advanced combinatorics, permanents, and Hafnians. The benchmark compares the mean, standard deviation, and skewness of the correlations across output-mode pairs, without reconstructing every output probability. These selective tests do not certify an arbitrary full output distribution.
Figure 1. Scheme of a boson sampling experiment. Photons are injected into the input ports, interact with a linear system and are detected at the output ports. We shall assume that the system is lossless and no “collisions” occur (namely photons are detected at different output ports).
Another photonic route, Gaussian boson sampling, employs squeezed inputs rather than fixed numbers of single photons. Hamilton et al. showed that its ideal, number-resolved output probabilities involve Hafnians , highlighting both the reach of photonic sampling and the importance of distinguishing the underlying matrix functions [25 ] .
In a recent large-scale experiment, Pan and collaborators realized Gaussian boson sampling with 1,024 squeezed states in 8,176 modes, registering events with up to 3,050 detector clicks and benchmarking the outputs against classical models [26 ] . This samples Gaussian outputs, rather than directly measuring individual permanents. For random interferometers, even the statistics of permanent-dependent outcomes pose subtle questions. Random permanents have their own mathematical history [27 ] and have not been systematically approached using the methods of statistical mechanics (a notable exception is [28 ] ).
Here we study the statistics of random permanents and Hafnians, rather than the evaluation of a prescribed instance.
Section II defines normalized moments for real and circular complex random matrices and derives the contour-integral representation of the permanent that will be used in the rest of the paper.
Section III turns the moments of real permanents into a replica calculation using Hubbard-Stratonovich fields. For Gaussian entries, this yields an auxiliary Hafnian representation; a large-N N N k ≥ 3 k \ge3 k ≥ 3 1 / N 1/N 1/ N
§ 2 treats complex Gaussian permanents, obtaining an exact moment–matrix-size duality and their large-N N N
In § 2.1 we use the moments to compute the full distribution of the normalized permanents, highlighting a transition region from the bulk (inaccessible to our methods) to the tail. In this transition region, a universal power law arises.
We compare numerical data for an ensemble of real matrices with the analytical calculation, and the agreement is remarkable. We remark that to access the distribution in the bulk, not just in the tail, one needs information about moments of order k ≤ 2 k \le2 k ≤ 2 [29 , 30 , 31 , 32 , 33 ] .
Section VI makes the connection to photonic experiments explicit: in the dilute, collision-free regime, singlephoton boson-sampling probabilities involve squared permanents of approximately Gaussian interferometer submatrices, whereas squeezed-state Gaussian boson sampling involves squared Hafnians. The moments describe the heavy tails of ideal output probabilities, compare the two sampling settings, and estimate the sample cost of cross-entropy and related statistical benchmarks. These ensemble predictions do not by themselves certify an individual device or establish the anti-concentration assumptions used in sampling-hardness arguments. Section VII summarizes the asymptotic laws, their universality, and their limitations.
The Appendix contains most of the technical derivations, including the saddle-point expansion, phase zero modes, fluctuation determinants, and finite-size corrections.
We believe the new techniques we introduce in this paper could spur interest in the problem of permanents of random matrices, which has been for many years the less-explored counterpart to determinants in random matrix theory, largely due to the historical lack of tractable analytical tools. Determinants are, in fact, expressed in terms of the matrix spectrum, and there is a large literature treating them [34 , 35 ] . Moreover, our uncovering of replica symmetry and the possibility of replica symmetry breaking connects to another important subfield of statistical physics: spin glasses and disordered systems [29 ] .
Permanents, Hafnians and their integral representations
We consider N × N N \times N N × N M M M P ( x ) P(x) P ( x ) x x x P ( x ) P(x) P ( x ) P ( e i θ x ) = P ( x ) P(e^{i\theta}x)=P(x) P ( e i θ x ) = P ( x ) ⟨ ⋅ ⟩ \langle\cdot\rangle ⟨ ⋅ ⟩ ∏ i j P ( M i j ) \prod_{ij}P(M_{ij}) ∏ ij P ( M ij )
For integer k k k per M \operatorname{per} M per M
⟨ ∣ per M ∣ 2 k ⟩ , \left\langle\lvert\operatorname{per} M\rvert^{2k}\right\rangle, ⟨ ∣ per M ∣ 2 k ⟩ , which we appropriately normalize to eliminate the factorials
⟨ ∣ per M ∣ 2 k ⟩ N ! k = ⟨ Y k ⟩ = ⟨ exp ( − k ( − ln ( Y ) ) ) ⟩ . (1) \frac{\left\langle\lvert\operatorname{per} M\rvert^{2k}\right\rangle}{N!^k}=\left\langle Y^k\right\rangle=\left\langle\exp\left(-k(-\ln(Y))\right)\right\rangle. \tag*{(1)} N ! k ⟨ ∣ per M ∣ 2 k ⟩ = ⟨ Y k ⟩ = ⟨ exp ( − k ( − ln ( Y )) ) ⟩ . (1) In this way ⟨ Y ⟩ = 1 \langle Y\rangle=1 ⟨ Y ⟩ = 1
Λ F ( k ) = lim N → ∞ 1 N ln ⟨ Y k ⟩ , (2) \Lambda_{\mathrm{F}}(k)=\lim_{N\to\infty}\frac{1}{N}\ln\langle Y^k\rangle, \tag*{(2)} Λ F ( k ) = N → ∞ lim N 1 ln ⟨ Y k ⟩ , (2) with the subscript F equal to R for real and to C for complex matrices, which plays the role of a free energy.
Notice that we are evaluating
∫ d M P ( M ) exp ( − k H ) , (3) \int dM P(M)\exp\left(-kH\right), \tag*{(3)} ∫ d M P ( M ) exp ( − k H ) , (3) namely the partition function of a system with inverse temperature k k k H = − ln ( Y ) ∝ N H=-\ln(Y)\propto N H = − ln ( Y ) ∝ N N → ∞ N\to\infty N → ∞
One ensemble relevant for the boson sampling problem [36 ] is that of complex matrices: the N × N N\times N N × N m × m m\times m m × m m \sqrt{m} m m m m N N N M i j M_{ij} M ij P ( z ) = 1 π e − ∣ z ∣ 2 P(z)=\frac{1}{\pi}e^{-\lvert z\rvert^2} P ( z ) = π 1 e − ∣ z ∣ 2 ⟨ ∣ M i j ∣ 2 ⟩ = 1 \langle\lvert M_{ij}\rvert^2\rangle=1 ⟨∣ M ij ∣ 2 ⟩ = 1 ⟨ M i j 2 ⟩ = 0 \langle M_{ij}^2\rangle=0 ⟨ M ij 2 ⟩ = 0 Λ \Lambda Λ C k C_k C k a 1 a_1 a 1 1 / N 1/N 1/ N
In order to average over M M M [28 ] . For every ρ > 0 \rho>0 ρ > 0
per ( M ) = ∏ i = 1 N ∮ ∣ u i ∣ = ρ d u i 2 π i u i 2 ∏ j = 1 N ∮ ∣ v j ∣ = ρ d v j 2 π i v j 2 exp ( ∑ i , j = 1 N u i M i j v j ) . (4) \operatorname{per}(M)=\prod_{i=1}^{N}\oint_{\lvert u_i\rvert=\rho}\frac{du_i}{2\pi i u_i^2}\prod_{j=1}^{N}\oint_{\lvert v_j\rvert=\rho}\frac{dv_j}{2\pi i v_j^2}\exp\left(\sum_{i,j=1}^{N}u_iM_{ij}v_j\right). \tag*{(4)} per ( M ) = i = 1 ∏ N ∮ ∣ u i ∣ = ρ 2 π i u i 2 d u i j = 1 ∏ N ∮ ∣ v j ∣ = ρ 2 π i v j 2 d v j exp ( i , j = 1 ∑ N u i M ij v j ) . (4) This representation is the twin of the very well-known representation of the determinant as an integral over Grassmann variables (as discussed in the Appendix)
det ( M ) = ∫ ∏ i = 1 N d ψ i d ψ ˉ i exp ( ∑ i , j = 1 N ψ ˉ i M i j ψ j ) , (5) \det(M)=\int\prod_{i=1}^{N}d\psi_i d\bar{\psi}_i\exp\left(\sum_{i,j=1}^{N}\bar{\psi}_iM_{ij}\psi_j\right), \tag*{(5)} det ( M ) = ∫ i = 1 ∏ N d ψ i d ψ ˉ i exp ( i , j = 1 ∑ N ψ ˉ i M ij ψ j ) , (5) which is widely used in the contexts of random matrix theory and Anderson localization [37 ] .
The proof of Eq. (4) is straightforward: in the expansion of the entire function e u T M v e^{u^TMv} e u T M v ( u T M v ) N (u^TMv)^N ( u T M v ) N M i 1 j 1 ⋯ M i N j N M_{i_1j_1}\cdots M_{i_Nj_N} M i 1 j 1 ⋯ M i N j N
For the Hafnian, we consider a 2 N × 2 N 2N\times2N 2 N × 2 N M M M M i j = M j i M_{ij}=M_{ji} M ij = M j i
hf ( M ) = ∑ T ∈ T 2 N ∏ { i , j } ∈ T M i j , (6) \operatorname{hf}(M)=\sum_{T\in\mathcal{T}_{2N}}\prod_{\{i,j\}\in T}M_{ij}, \tag*{(6)} hf ( M ) = T ∈ T 2 N ∑ { i , j } ∈ T ∏ M ij , (6) where T 2 N \mathcal{T}_{2N} T 2 N ( 2 N − 1 ) ! (2N-1)! ( 2 N − 1 )! T T T [ 2 N ] = { 1 , … , 2 N } [2N]=\{1,\ldots,2N\} [ 2 N ] = { 1 , … , 2 N }
The Hafnian has a contour representation analogous to (4), with a single set of variables, since rows and columns coincide:
hf ( M ) = ∏ i = 1 2 N ∮ ∣ u i ∣ = ρ d u i 2 π i u i 2 exp ( ∑ i < j u i M i j u j ) . (7) \operatorname{hf}(M)=\prod_{i=1}^{2N}\oint_{|u_i|=\rho}\frac{du_i}{2\pi i u_i^2}\exp\left(\sum_{i<j}u_iM_{ij}u_j\right). \tag*{(7)} hf ( M ) = i = 1 ∏ 2 N ∮ ∣ u i ∣ = ρ 2 π i u i 2 d u i exp ( i < j ∑ u i M ij u j ) . (7) The proof is the same as for (4): in ( ∑ i < j M i j u i u j ) N / N ! \left(\sum_{i<j}M_{ij}u_i u_j\right)^N/N! ( ∑ i < j M ij u i u j ) N / N ! N N N [ 2 N ] [2N] [ 2 N ] N ! N! N !
These representations, with the matrix in the exponent, turn the problem of computing the moments of a permanent or a Hafnian into a form amenable to treatments using the full arsenal of statistical mechanics of disordered systems.
Real matrices
To compute ⟨ ∣ per ( M ) ∣ 2 k ⟩ = ⟨ ( per M ) 2 k ⟩ \langle|\operatorname{per}(M)|^{2k}\rangle=\langle(\operatorname{per}M)^{2k}\rangle ⟨ ∣ per ( M ) ∣ 2 k ⟩ = ⟨( per M ) 2 k ⟩ 2 k × N 2k\times N 2 k × N u u u 2 k × N 2k\times N 2 k × N v v v
d μ 2 k ( u , v ) = ∏ a = 1 2 k ∏ i = 1 N d u i a 2 π i ( u i a ) 2 ∏ a = 1 2 k ∏ j = 1 N d v j a 2 π i ( v j a ) 2 . (8) d\mu_{2k}(u,v)=\prod_{a=1}^{2k}\prod_{i=1}^{N}\frac{du_i^a}{2\pi i (u_i^a)^2}\prod_{a=1}^{2k}\prod_{j=1}^{N}\frac{dv_j^a}{2\pi i (v_j^a)^2}. \tag*{(8)} d μ 2 k ( u , v ) = a = 1 ∏ 2 k i = 1 ∏ N 2 π i ( u i a ) 2 d u i a a = 1 ∏ 2 k j = 1 ∏ N 2 π i ( v j a ) 2 d v j a . (8) With this notation
( per M ) 2 k = ∮ d μ 2 k ( u , v ) exp ( ∑ i , j M i j X i j ) , (\operatorname{per}M)^{2k}=\oint d\mu_{2k}(u,v)\exp\left(\sum_{i,j}M_{ij}X_{ij}\right), ( per M ) 2 k = ∮ d μ 2 k ( u , v ) exp ( i , j ∑ M ij X ij ) , X i j = ∑ a = 1 2 k u i a v j a . (9) X_{ij}=\sum_{a=1}^{2k}u_i^av_j^a. \tag*{(9)} X ij = a = 1 ∑ 2 k u i a v j a . (9) We assume the distribution of M i j M_{ij} M ij P ( x ) = P ( − x ) P(x)=P(-x) P ( x ) = P ( − x ) κ 2 = 1 \kappa_2=1 κ 2 = 1 κ m \kappa_m κ m m m m
ψ ( y ) = ln ⟨ e y X ⟩ = ln ( ∫ d x P ( x ) e y x ) = ∑ m = 1 ∞ κ m m ! y m . (10) \psi(y)=\ln\langle e^{yX}\rangle=\ln\left(\int dxP(x)e^{yx}\right)=\sum_{m=1}^{\infty}\frac{\kappa_m}{m!}y^m. \tag*{(10)} ψ ( y ) = ln ⟨ e y X ⟩ = ln ( ∫ d x P ( x ) e y x ) = m = 1 ∑ ∞ m ! κ m y m . (10) Cumulants can be seen also as connected correlation functions, in particular κ 4 = κ 2 2 ( K − 3 ) \kappa_4=\kappa_2^2(K-3) κ 4 = κ 2 2 ( K − 3 ) K K K κ 2 = 1 \kappa_2=1 κ 2 = 1 κ m ≥ 3 = 0 \kappa_{m\geq3}=0 κ m ≥ 3 = 0 X = ± 1 X=\pm1 X = ± 1 ψ ( y ) = ln ( cosh ( y ) ) \psi(y)=\ln(\cosh(y)) ψ ( y ) = ln ( cosh ( y ))
Now, by ⟨ e M i j X i j ⟩ = e ψ ( X i j ) \langle e^{M_{ij}X_{ij}}\rangle=e^{\psi(X_{ij})} ⟨ e M ij X ij ⟩ = e ψ ( X ij )
⟨ ( per M ) 2 k ⟩ = ∮ d μ 2 k ( u , v ) exp ( ∑ i , j = 1 N ψ ( X i j ) ) = ∮ d μ 2 k ( u , v ) exp ( ∑ i , j = 1 N ∑ m = 1 ∞ κ m m ! X i j m ) . (11) \begin{aligned}
\left\langle(\operatorname{per}M)^{2k}\right\rangle=\oint d\mu_{2k}(u,v)\exp\left(\sum_{i,j=1}^{N}\psi(X_{ij})\right) \\
&=\oint d\mu_{2k}(u,v)\exp\left(\sum_{i,j=1}^{N}\sum_{m=1}^{\infty}\frac{\kappa_m}{m!}X_{ij}^m\right). \tag*{(11)}
\end{aligned} ⟨ ( per M ) 2 k ⟩ = ∮ d μ 2 k ( u , v ) exp ( i , j = 1 ∑ N ψ ( X ij ) ) = ∮ d μ 2 k ( u , v ) exp ( i , j = 1 ∑ N m = 1 ∑ ∞ m ! κ m X ij m ) . (11) The contour integral (9) extracts the coefficient of the monomial in which every u i a u_i^a u i a v j a v_j^a v j a
one. Any monomial divisible by some ( u i a ) 2 (u_i^a)^2 ( u i a ) 2 ( v j a ) 2 (v_j^a)^2 ( v j a ) 2 R R R e A + R e^{A+R} e A + R e A e^A e A
X i j m m ! = 1 m ! ∑ a 1 , … , a m ∏ l = 1 m u i a l v j a l (12) \frac{X_{ij}^m}{m!}=\frac{1}{m!}\sum_{a_1,\ldots,a_m}\prod_{l=1}^{m}u_i^{a_l}v_j^{a_l} \tag*{(12)} m ! X ij m = m ! 1 a 1 , … , a m ∑ l = 1 ∏ m u i a l v j a l (12) the terms with a repeated replica index are of this kind, and each subset B B B m m m m ! m! m !
X i j m m ! ⟶ ∑ B ⊆ [ 2 k ] ∣ B ∣ = m u i B v j B . (13) \frac{X_{ij}^m}{m!}\longrightarrow\sum_{\substack{B\subseteq[2k]\\ |B|=m}}u_i^Bv_j^B. \tag*{(13)} m ! X ij m ⟶ B ⊆ [ 2 k ] ∣ B ∣ = m ∑ u i B v j B . (13) where [ 2 k ] = { 1 , … , 2 k } [2k]=\{1,\ldots,2k\} [ 2 k ] = { 1 , … , 2 k } m > 2 k m>2k m > 2 k m m m
⟨ ( per M ) 2 k ⟩ = ∮ d μ 2 k ( u , v ) exp ( ∑ B ∈ B 2 k κ ∣ B ∣ R B S B ) , \left\langle(\operatorname{per}M)^{2k}\right\rangle=\oint d\mu_{2k}(u,v)\exp\left(\sum_{B\in\mathcal{B}_{2k}}\kappa_{|B|}R_BS_B\right), ⟨ ( per M ) 2 k ⟩ = ∮ d μ 2 k ( u , v ) exp ( B ∈ B 2 k ∑ κ ∣ B ∣ R B S B ) , R B = ∑ i = 1 N u i B , S B = ∑ j = 1 N v j B . (14) R_B=\sum_{i=1}^{N}u_i^B,\qquad S_B=\sum_{j=1}^{N}v_j^B. \tag*{(14)} R B = i = 1 ∑ N u i B , S B = j = 1 ∑ N v j B . (14) Here,
B 2 k : = { B ⊆ [ 2 k ] : ∣ B ∣ ∈ { 2 , 4 , … , 2 k } } , (15) \mathcal{B}_{2k}:=\{B\subseteq[2k]:|B|\in\{2,4,\ldots,2k\}\}, \tag*{(15)} B 2 k := { B ⊆ [ 2 k ] : ∣ B ∣ ∈ { 2 , 4 , … , 2 k }} , (15) denotes the family of nonempty even-cardinality subsets, and for every subset B ⊆ [ 2 k ] B\subseteq[2k] B ⊆ [ 2 k ]
u i B : = ∏ a ∈ B u i a , v j B : = ∏ a ∈ B v j a . (16) u_i^B:=\prod_{a\in B}u_i^a,\qquad v_j^B:=\prod_{a\in B}v_j^a. \tag*{(16)} u i B := a ∈ B ∏ u i a , v j B := a ∈ B ∏ v j a . (16) R B R_B R B S B S_B S B B ⊆ [ 2 k ] B\subseteq[2k] B ⊆ [ 2 k ] B B B ∣ B ∣ = 2 |B|=2 ∣ B ∣ = 2 R a b = ∑ i u i a u i b R_{ab}=\sum_i u_i^a u_i^b R ab = ∑ i u i a u i b ∣ B ∣ ≥ 4 |B|\geq4 ∣ B ∣ ≥ 4 κ m \kappa_m κ m m ≥ 4 m\geq4 m ≥ 4
Hubbard-Stratonovich decoupling
We now introduce a complex Hubbard-Stratonovich (HS) integral. This is a standard trick to rewrite the average moments, and therefore the quenched average, of fully-connected statistical mechanics models like the SK model [38 ] . In the context of Anderson localization (determinants) this leads to the nonlinear sigma model [37 ] and in the context of random permanents it was used before in [28 ] .
Introducing a new complex variable by defining an average
E z ( ⋅ ) = ∫ d 2 z π e − ∣ z ∣ 2 ( ⋅ ) . (17) \mathbb{E}_z(\cdot)=\int\frac{d^2z}{\pi}e^{-|z|^2}(\cdot). \tag*{(17)} E z ( ⋅ ) = ∫ π d 2 z e − ∣ z ∣ 2 ( ⋅ ) . (17) for an arbitrary number of complex variables z z z B B B
E z e κ z R + z ˉ S = e κ R S , (18) \mathbb{E}_{z} e^{\kappa zR+\bar{z}S}=e^{\kappa RS}, \tag*{(18)} E z e κ z R + z ˉ S = e κ R S , (18) where κ z R ≡ ∑ B κ ∣ B ∣ z B R B \kappa zR\equiv\sum_{B}\kappa^{|B|}z_{B}R_{B} κ z R ≡ ∑ B κ ∣ B ∣ z B R B z ˉ S ≡ ∑ B z ˉ B S B \bar{z}S\equiv\sum_{B}\bar{z}_{B}S_{B} z ˉ S ≡ ∑ B z ˉ B S B B ∈ B 2 k B\in\mathcal{B}_{2k} B ∈ B 2 k
⟨ ( per M ) 2 k ⟩ = E z ∮ d μ 2 k ( u , v ) exp ( ∑ i = 1 N ∑ B ∈ B 2 k κ ∣ B ∣ z B u i B ) \left\langle(\operatorname{per}M)^{2k}\right\rangle=\mathbb{E}_{z}\oint d\mu_{2k}(u,v)
\exp\left(\sum_{i=1}^{N}\sum_{B\in\mathcal{B}_{2k}}\kappa_{|B|}z_{B}u_{i}^{B}\right) ⟨ ( per M ) 2 k ⟩ = E z ∮ d μ 2 k ( u , v ) exp ( i = 1 ∑ N B ∈ B 2 k ∑ κ ∣ B ∣ z B u i B ) × exp ( ∑ j = 1 N ∑ B ∈ B 2 k z ˉ B v j B ) . (19) {}\times\exp\left(\sum_{j=1}^{N}\sum_{B\in\mathcal{B}_{2k}}\bar{z}_{B}v_{j}^{B}\right). \tag*{(19)} × exp ( j = 1 ∑ N B ∈ B 2 k ∑ z ˉ B v j B ) . (19) In this way, the rows and columns have decoupled, as the sites do in the replica calculation for spin glasses: the integrand is a product over i i i ( u a i ) a (u_{a}^{i})_{a} ( u a i ) a j j j ( v a j ) a (v_{a}^{j})_{a} ( v a j ) a
Now let us go back to (14) for the case m = 2 m=2 m = 2 B = { a , b } B=\{a,b\} B = { a , b } z B = z a b z_{B}=z_{ab} z B = z ab R B = R a b R_{B}=R_{ab} R B = R ab S B = S a b S_{B}=S_{ab} S B = S ab z a b ∼ Q a b z_{ab}\sim Q_{ab} z ab ∼ Q ab Q a b c d ∼ z a b c d Q_{abcd}\sim z_{abcd} Q ab c d ∼ z ab c d
We can do the integrals over d μ 2 k ( u , v ) d\mu_{2k}(u,v) d μ 2 k ( u , v )
∏ a = 1 2 k ∮ ∣ u a ∣ = ρ d u a 2 π i ( u a ) 2 exp ( ∑ B ∈ B 2 k c B u B ) = ∑ π ∈ Π 2 k e v ∏ B ∈ π c B , (20) \prod_{a=1}^{2k}\oint_{|u^{a}|=\rho}\frac{du^{a}}{2\pi i(u^{a})^{2}}\exp\left(\sum_{B\in\mathcal{B}_{2k}}c_{B}u^{B}\right)
=\sum_{\pi\in\Pi_{2k}^{\mathrm{ev}}}\prod_{B\in\pi}c_{B}, \tag*{(20)} a = 1 ∏ 2 k ∮ ∣ u a ∣ = ρ 2 π i ( u a ) 2 d u a exp ( B ∈ B 2 k ∑ c B u B ) = π ∈ Π 2 k ev ∑ B ∈ π ∏ c B , (20) for any coefficients c B c_{B} c B Π 2 k e v \Pi_{2k}^{\mathrm{ev}} Π 2 k ev [ 2 k ] [2k] [ 2 k ] π ∈ Π 2 k e v \pi\in\Pi_{2k}^{\mathrm{ev}} π ∈ Π 2 k ev B ⊆ [ 2 k ] B\subseteq[2k] B ⊆ [ 2 k ] ∣ B ∣ |B| ∣ B ∣ ≥ 2 \geq2 ≥ 2 [ 2 k ] [2k] [ 2 k ] ( ∑ B c B u B ) n / n ! (\sum_{B}c_{B}u^{B})^{n}/n! ( ∑ B c B u B ) n / n ! u 1 , … , u 2 k u^{1},\ldots,u^{2k} u 1 , … , u 2 k n n n [ 2 k ] [2k] [ 2 k ] n ! n! n !
F κ ( z ) = ∑ π ∈ Π 2 k e v ∏ B ∈ π κ ∣ B ∣ z B , G ( z ) = ∑ π ∈ Π 2 k e v ∏ B ∈ π z B , (21) F_{\kappa}(z)=\sum_{\pi\in\Pi_{2k}^{\mathrm{ev}}}\prod_{B\in\pi}\kappa_{|B|}z_{B},\qquad
G(z)=\sum_{\pi\in\Pi_{2k}^{\mathrm{ev}}}\prod_{B\in\pi}z_{B}, \tag*{(21)} F κ ( z ) = π ∈ Π 2 k ev ∑ B ∈ π ∏ κ ∣ B ∣ z B , G ( z ) = π ∈ Π 2 k ev ∑ B ∈ π ∏ z B , (21) Each row contributes F κ ( z ) F_{\kappa}(z) F κ ( z ) G ( z ˉ ) = G ( z ) ‾ G(\bar{z})=\overline{G(z)} G ( z ˉ ) = G ( z ) G G G
Finally, we reach the HS form with one family of z B z_{B} z B ∣ B ∣ = m |B|=m ∣ B ∣ = m κ m \kappa_{m} κ m
⟨ ( per M ) 2 k ⟩ = E z [ F κ ( z ) N G ( z ) ‾ N ] . (22) \left\langle(\operatorname{per}M)^{2k}\right\rangle=\mathbb{E}_{z}\left[F_{\kappa}(z)^{N}\overline{G(z)}^{N}\right]. \tag*{(22)} ⟨ ( per M ) 2 k ⟩ = E z [ F κ ( z ) N G ( z ) N ] . (22) The Gaussian distribution
With this formal machinery, we will now solve the Gaussian case first. For Gaussian entries κ 2 = 1 \kappa_{2}=1 κ 2 = 1 κ m = 0 \kappa_{m}=0 κ m = 0 m ≥ 4 m\geq4 m ≥ 4 ∣ B ∣ ≥ 4 |B|\geq4 ∣ B ∣ ≥ 4 T ∈ T 2 k T\in\mathcal{T}_{2k} T ∈ T 2 k [ 2 k ] [2k] [ 2 k ]
F κ ( z ) = ∑ T ∈ T 2 k ∏ { a , b } ∈ T z a b = hf ( z ) , (23) F_{\kappa}(z)=\sum_{T\in\mathcal{T}_{2k}}\prod_{\{a,b\}\in T}z_{ab}=\operatorname{hf}(z), \tag*{(23)} F κ ( z ) = T ∈ T 2 k ∑ { a , b } ∈ T ∏ z ab = hf ( z ) , (23) the Hafnian of the HS fields matrix ( z a b ) a < b (z_{ab})_{a<b} ( z ab ) a < b G ( z ) G(z) G ( z ) z B z_{B} z B ∣ B ∣ = 2 |B|=2 ∣ B ∣ = 2 E z \mathbb{E}_{z} E z G ( z ) = hf ( z ) G(z)=\operatorname{hf}(z) G ( z ) = hf ( z )
⟨ ( per M ) 2 k ⟩ = E z ∣ hf ( z ) ∣ 2 N , (24) \left\langle(\operatorname{per}M)^{2k}\right\rangle=\mathbb{E}_{z}|\operatorname{hf}(z)|^{2N}, \tag*{(24)} ⟨ ( per M ) 2 k ⟩ = E z ∣ hf ( z ) ∣ 2 N , (24) only over the variables z B z_{B} z B B B B [ 2 k ] [2k] [ 2 k ] D = k ( 2 k − 1 ) D=k(2k-1) D = k ( 2 k − 1 ) z B = y B N z_{B}=y_{B}\sqrt{N} z B = y B N
⟨ ( per M ) 2 k ⟩ = N k N + D ∫ ∏ e = 1 D d 2 y e π exp [ − N Φ R ( y ) ] . (25) \left\langle(\operatorname{per}M)^{2k}\right\rangle=N^{kN+D}\int\prod_{e=1}^{D}\frac{d^{2}y_{e}}{\pi}\exp[-N\Phi_{\mathbb{R}}(y)]. \tag*{(25)} ⟨ ( per M ) 2 k ⟩ = N k N + D ∫ e = 1 ∏ D π d 2 y e exp [ − N Φ R ( y )] . (25) where
Φ R = ∑ e = 1 D ∣ y e ∣ 2 − ln hf ( y ) − ln hf ( y ) ‾ . (26) \Phi_{\mathbb{R}}=\sum_{e=1}^{D}|y_{e}|^{2}-\ln\operatorname{hf}(y)-\ln\overline{\operatorname{hf}(y)}. \tag*{(26)} Φ R = e = 1 ∑ D ∣ y e ∣ 2 − ln hf ( y ) − ln hf ( y ) . (26) We can now compute the integral using the saddle point method. For large N N N ∂ y e Φ R = 0 \partial_{y_{e}}\Phi_{\mathbb{R}}=0 ∂ y e Φ R = 0
y e = ( ∂ e hf ( y ) hf ( y ) ) . (27) y_{e}=\left(\frac{\partial_{e}\operatorname{hf}(y)}{\operatorname{hf}(y)}\right). \tag*{(27)} y e = ( hf ( y ) ∂ e hf ( y ) ) . (27) Multiplying by y ˉ e \bar{y}_{e} y ˉ e e = 1 , … , D e=1,\ldots,D e = 1 , … , D ∑ e = 1 D y e ∂ e hf = k hf \sum_{e=1}^{D}y_{e}\partial_{e}\operatorname{hf}=k\operatorname{hf} ∑ e = 1 D y e ∂ e hf = k hf ∑ e = 1 D ∣ y e ∣ 2 = k \sum_{e=1}^{D}|y_{e}|^{2}=k ∑ e = 1 D ∣ y e ∣ 2 = k
There is a replica-symmetric (RS) saddle y e = y 0 y_{e}=y_{0} y e = y 0 e = 1 , … , D e=1,\ldots,D e = 1 , … , D [39 ] that indeed, for k ≥ 3 k\geq3 k ≥ 3 ∂ e hf = ( 2 k − 3 ) ! ! y 0 k − 1 \partial_{e}\operatorname{hf}=(2k-3)!!\,y_{0}^{k-1} ∂ e hf = ( 2 k − 3 )!! y 0 k − 1 hf = ( 2 k − 1 ) ! ! y 0 k \operatorname{hf}=(2k-1)!!\,y_{0}^{k} hf = ( 2 k − 1 )!! y 0 k y 0 = 1 / ( ( 2 k − 1 ) y 0 ) y_{0}=1/((2k-1)y_{0}) y 0 = 1/ (( 2 k − 1 ) y 0 )
y 0 2 = 1 2 k − 1 = q 0 . (28) y_{0}^{2}=\frac{1}{2k-1}=q_{0}. \tag*{(28)} y 0 2 = 2 k − 1 1 = q 0 . (28) The saddle value of ∣ y e ∣ 2 |y_{e}|^{2} ∣ y e ∣ 2 ∣ y e ∣ 2 = q e |y_{e}|^{2}=q_{e} ∣ y e ∣ 2 = q e ∣ z e ∣ 2 ≈ N q e |z_{e}|^{2}\approx Nq_{e} ∣ z e ∣ 2 ≈ N q e z e z_{e} z e a a a b b b y s = ( y 0 , … , y 0 ) y_s=(y_0,\ldots,y_0) y s = ( y 0 , … , y 0 ) N N N
Λ R ( k ) = k − Φ R ( y s ) = 2 k ln 2 + 2 ln Γ ( k + 1 2 ) − ln π − k ln ( 2 k − 1 ) , (29) \Lambda_{\mathrm{R}}(k)=k-\Phi_{\mathrm{R}}(y_s)
=2k\ln2+2\ln\Gamma\left(k+\frac{1}{2}\right)-\ln\pi-k\ln(2k-1), \tag*{(29)} Λ R ( k ) = k − Φ R ( y s ) = 2 k ln 2 + 2 ln Γ ( k + 2 1 ) − ln π − k ln ( 2 k − 1 ) , (29) it is an analytic function of k > 2 k>2 k > 2 N N N
lim N → ∞ 1 N ln ⟨ Y k ⟩ = Λ R ( k ) . \lim_{N\to\infty}\frac{1}{N}\ln\left\langle Y^k\right\rangle=\Lambda_{\mathrm{R}}(k). N → ∞ lim N 1 ln ⟨ Y k ⟩ = Λ R ( k ) . The polynomial prefactor and the 1 / N 1/N 1/ N
⟨ Y k ⟩ = C k R ( 1 + a 1 R N + O ( 1 N 2 ) ) N k / 2 e N Λ R ( k ) , (30) \left\langle Y^k\right\rangle=C_k^{\mathrm{R}}\left(1+\frac{a_1^{\mathrm{R}}}{N}+O\left(\frac{1}{N^2}\right)\right)N^{k/2}e^{N\Lambda_{\mathrm{R}}(k)}, \tag*{(30)} ⟨ Y k ⟩ = C k R ( 1 + N a 1 R + O ( N 2 1 ) ) N k /2 e N Λ R ( k ) , (30) where
C k R = 2 k − 1 ( 2 π ) k / 2 ( k − 1 2 k − 1 ) k − 1 2 ( 1 + 1 4 ( k − 1 ) ( k − 2 ) ) k ( 2 k − 3 ) / 2 , (31) C_k^{\mathrm{R}}=2^{k-1}(2\pi)^{k/2}\left(\frac{k-1}{2k-1}\right)^{k-\frac{1}{2}}\left(1+\frac{1}{4(k-1)(k-2)}\right)^{k(2k-3)/2}, \tag*{(31)} C k R = 2 k − 1 ( 2 π ) k /2 ( 2 k − 1 k − 1 ) k − 2 1 ( 1 + 4 ( k − 1 ) ( k − 2 ) 1 ) k ( 2 k − 3 ) /2 , (31) and
a 1 R ( k ) = k ( 8 k 4 − 44 k 3 + 124 k 2 − 178 k + 93 ) 48 ( k − 1 ) ( k − 2 ) 2 ( 2 k − 5 ) . (32) a_1^{\mathrm{R}}(k)=\frac{k\left(8k^4-44k^3+124k^2-178k+93\right)}{48(k-1)(k-2)^2(2k-5)}. \tag*{(32)} a 1 R ( k ) = 48 ( k − 1 ) ( k − 2 ) 2 ( 2 k − 5 ) k ( 8 k 4 − 44 k 3 + 124 k 2 − 178 k + 93 ) . (32) For large k k k a 1 R ∼ k / 12 a_1^{\mathrm{R}}\sim k/12 a 1 R ∼ k /12 k > 2 k>2 k > 2 N ≫ k / 12 N\gg k/12 N ≫ k /12 k = 3 k=3 k = 3 N = 9 N=9 N = 9 ∼ 29 % \sim29\% ∼ 29% 3 % 3\% 3% 1 / N 1/N 1/ N
General distributions
We start this section by briefly showing that Λ R ( k ) \Lambda_{\mathrm{R}}(k) Λ R ( k ) P ( x ) P(x) P ( x ) x x x ⟨ Y k ⟩ \left\langle Y^k\right\rangle ⟨ Y k ⟩ C k R C_k^{\mathrm{R}} C k R e W k e^{W_k} e W k κ 4 \kappa_4 κ 4 1 / N 1/N 1/ N
The idea is that at the RS saddle only the pair fields are macroscopic, ∣ z e ∣ 2 ≈ N q 0 |z_e|^2\approx Nq_0 ∣ z e ∣ 2 ≈ N q 0 z B z_B z B ∣ B ∣ ≥ 4 |B|\geq4 ∣ B ∣ ≥ 4 z e = N y e z_e=\sqrt{N}y_e z e = N y e e = 1 , … , D e=1,\ldots,D e = 1 , … , D π ∈ Π 2 k e v \pi\in\Pi_{2k}^{\mathrm{ev}} π ∈ Π 2 k ev p p p F κ F_\kappa F κ N p / 2 N^{p/2} N p /2 p = k p=k p = k p = k − 2 p=k-2 p = k − 2 1 / N 1/N 1/ N N − 3 / 2 N^{-3/2} N − 3/2
F κ ( z ) = N k / 2 hf ( y ) [ 1 + κ 4 N ∑ B ⊆ [ 2 k ] ∣ B ∣ = 4 z B h B ( y ) hf ( y ) + O ( N − 3 / 2 ) ] , (33) F_\kappa(z)=N^{k/2}\operatorname{hf}(y)\left[1+\frac{\kappa_4}{N}\sum_{\substack{B\subseteq[2k]\\ |B|=4}}z_B\frac{h_B(y)}{\operatorname{hf}(y)}+O\left(N^{-3/2}\right)\right], \tag*{(33)} F κ ( z ) = N k /2 hf ( y ) 1 + N κ 4 B ⊆ [ 2 k ] ∣ B ∣ = 4 ∑ z B hf ( y ) h B ( y ) + O ( N − 3/2 ) , (33) where h B ( y ) h_B(y) h B ( y ) y a b y_{ab} y ab a , b ∈ [ 2 k ] ∖ B a,b\in[2k]\setminus B a , b ∈ [ 2 k ] ∖ B G ( z ˉ ) G(\bar z) G ( z ˉ ) κ 4 \kappa_4 κ 4 1 1 1 z B z_B z B z ˉ B \bar z_B z ˉ B N N N
⟨ ( per M ) 2 k ⟩ = N k N + D ∫ ∏ e = 1 D d 2 y e π e − N Φ R ( y ) × exp ( κ 4 ∑ B ⊆ [ 2 k ] ∣ B ∣ = 4 ∣ h B ( y ) hf ( y ) ∣ 2 ) × [ 1 + O ( 1 / N ) ] . (34) \left\langle(\operatorname{per}M)^{2k}\right\rangle=N^{kN+D}\int\prod_{e=1}^{D}\frac{d^2y_e}{\pi}e^{-N\Phi_{\mathrm{R}}(y)}
\times\exp\left(\kappa_4\sum_{\substack{B\subseteq[2k]\\ |B|=4}}\left|\frac{h_B(y)}{\operatorname{hf}(y)}\right|^2\right)
\times\left[1+O(1/N)\right]. \tag*{(34)} ⟨ ( per M ) 2 k ⟩ = N k N + D ∫ e = 1 ∏ D π d 2 y e e − N Φ R ( y ) × exp κ 4 B ⊆ [ 2 k ] ∣ B ∣ = 4 ∑ hf ( y ) h B ( y ) 2 × [ 1 + O ( 1/ N ) ] . (34) The new factor is of order one, so it does not move the saddle point and Λ R ( k ) \Lambda_{\mathrm{R}}(k) Λ R ( k )
This is the proof that the rate Λ R ( k ) \Lambda_{\mathrm{R}}(k) Λ R ( k ) h B ( y s ) = ( 2 k − 5 ) ! ! y 0 k − 2 h_B(y_s)=(2k-5)!!\,y_0^{k-2} h B ( y s ) = ( 2 k − 5 )!! y 0 k − 2
h B ( y s ) hf ( y s ) = 1 ( 2 k − 1 ) ( 2 k − 3 ) y 0 2 = 1 2 k − 3 (35) \frac{h_B(y_s)}{\operatorname{hf}(y_s)}=\frac{1}{(2k-1)(2k-3)y_0^2}=\frac{1}{2k-3} \tag*{(35)} hf ( y s ) h B ( y s ) = ( 2 k − 1 ) ( 2 k − 3 ) y 0 2 1 = 2 k − 3 1 (35) for each of the ( 2 k 4 ) \binom{2k}{4} ( 4 2 k ) C k R → e W k C k R C_k^{\mathrm{R}}\to e^{W_k}C_k^{\mathrm{R}} C k R → e W k C k R
W k = κ 4 ( 2 k 4 ) 1 ( 2 k − 3 ) 2 = κ 4 k ( k − 1 ) ( 2 k − 1 ) 6 ( 2 k − 3 ) . (36) W_k=\kappa_4\binom{2k}{4}\frac{1}{(2k-3)^2}=\kappa_4\frac{k(k-1)(2k-1)}{6(2k-3)}. \tag*{(36)} W k = κ 4 ( 4 2 k ) ( 2 k − 3 ) 2 1 = κ 4 6 ( 2 k − 3 ) k ( k − 1 ) ( 2 k − 1 ) . (36) Taking as an example Rademacher M i j = ± 1 M_{ij}=\pm1 M ij = ± 1 κ 2 = 1 \kappa_2=1 κ 2 = 1 κ 4 = − 2 \kappa_4=-2 κ 4 = − 2 κ 6 = 16 \kappa_6=16 κ 6 = 16
W k ± 1 = − k ( k − 1 ) ( 2 k − 1 ) 3 ( 2 k − 3 ) , (37) W_k^{\pm1}=-\frac{k(k-1)(2k-1)}{3(2k-3)}, \tag*{(37)} W k ± 1 = − 3 ( 2 k − 3 ) k ( k − 1 ) ( 2 k − 1 ) , (37) while the large-N N N e W k C k R e^{W_k}C_k^{\mathrm{R}} e W k C k R k = 3 , 4 , 5 k=3,4,5 k = 3 , 4 , 5 k = 2 k=2 k = 2 e W 2 = e − 2 e^{W_2}=e^{-2} e W 2 = e − 2 ± 1 \pm1 ± 1 N N N κ 6 = 16 \kappa_6=16 κ 6 = 16 1 / N 1/N 1/ N ± 1 \pm1 ± 1 1 / N 1/N 1/ N
Dividing ⟨ ( per M ) 2 k ⟩ \left\langle(\operatorname{per}M)^{2k}\right\rangle ⟨ ( per M ) 2 k ⟩ N k N^k N k ⟨ Y k ⟩ \left\langle Y^k\right\rangle ⟨ Y k ⟩ O ( 1 / N ) O(1/N) O ( 1/ N )
Table 1. Moments ⟨ Y k ⟩ = ⟨ ( per M ) 2 k ⟩ / N ! \langle Y^k\rangle= \langle(\operatorname{per} M)^{2k}\rangle/N! ⟨ Y k ⟩ = ⟨( per M ) 2 k ⟩ / N ! C k R N k / 2 e N Λ R ( k ) C_k^{\mathrm{R}}N^{k/2}e^{N\Lambda_{\mathrm{R}}(k)} C k R N k /2 e N Λ R ( k ) 1 + a 1 R / N 1+a_1^{\mathrm{R}}/N 1 + a 1 R / N
⟨ Y k ⟩ = e W k C k R N k / 2 e N Λ R ( k ) [ 1 + a 1 R ( k ) + w 1 ( k ) N + O ( N − 2 ) ] , (38) \langle Y^k\rangle=e^{W_k}C_k^{\mathrm{R}}N^{k/2}e^{N\Lambda_{\mathrm{R}}(k)}
\left[1+\frac{a_1^{\mathrm{R}}(k)+w_1(k)}{N}+O(N^{-2})\right], \tag*{(38)} ⟨ Y k ⟩ = e W k C k R N k /2 e N Λ R ( k ) [ 1 + N a 1 R ( k ) + w 1 ( k ) + O ( N − 2 ) ] , (38) w 1 ( k ) = α ( k ) κ 4 + β ( k ) κ 4 2 + γ ( k ) κ 6 , w_1(k)=\alpha(k)\kappa_4+\beta(k)\kappa_4^2+\gamma(k)\kappa_6, w 1 ( k ) = α ( k ) κ 4 + β ( k ) κ 4 2 + γ ( k ) κ 6 , with a 1 R a_1^{\mathrm{R}} a 1 R
α = k ( 2 k − 1 ) 2 6 ( 2 k − 3 ) ( 2 k − 5 ) , \alpha=\frac{k(2k-1)^2}{6(2k-3)(2k-5)}, α = 6 ( 2 k − 3 ) ( 2 k − 5 ) k ( 2 k − 1 ) 2 , β = − k ( k − 1 ) ( k − 2 ) ( 2 k − 1 ) 2 18 ( 2 k − 3 ) 2 , (39) \beta=-\frac{k(k-1)(k-2)(2k-1)^2}{18(2k-3)^2}, \tag*{(39)} β = − 18 ( 2 k − 3 ) 2 k ( k − 1 ) ( k − 2 ) ( 2 k − 1 ) 2 , (39) γ = k ( k − 1 ) ( k − 2 ) ( 2 k − 1 ) 2 90 ( 2 k − 3 ) ( 2 k − 5 ) , \gamma=\frac{k(k-1)(k-2)(2k-1)^2}{90(2k-3)(2k-5)}, γ = 90 ( 2 k − 3 ) ( 2 k − 5 ) k ( k − 1 ) ( k − 2 ) ( 2 k − 1 ) 2 , while κ 8 \kappa_8 κ 8 N − 2 N^{-2} N − 2 Figure 2 .
Figure 2. Rate function Λ R ( k ) \Lambda_{\mathrm{R}}(k) Λ R ( k ) Λ R = 0 \Lambda_{\mathrm{R}}=0 Λ R = 0 k ≤ 2 k\leq2 k ≤ 2 k > 2 k>2 k > 2 k < 2 k<2 k < 2 ln ⟨ Y k ⟩ N + 1 / ⟨ Y k ⟩ N = 1 2 ln N + 1 N \ln\langle Y^k\rangle_{N+1}/\langle Y^k\rangle_N=\frac{1}{2}\ln\frac{N+1}{N} ln ⟨ Y k ⟩ N + 1 / ⟨ Y k ⟩ N = 2 1 ln N N + 1 k > 2 k>2 k > 2 ± 1 \pm1 ± 1 N = 8 , 9 N=8,9 N = 8 , 9 N = 12 , 13 N=12,13 N = 12 , 13
Complex Gaussian matrices
In this section we compute the moments of permanents and of Hafnians for the complex, circular ensemble of the M i j M_{ij} M ij
Permanent
The integral representation has the same 2 k × N 2k \times N 2 k × N u i a u_i^a u i a v j a v_j^a v j a d μ 2 k ( u , v ) d\mu_{2k}(u,v) d μ 2 k ( u , v ) a ∈ [ k ] a \in[k] a ∈ [ k ] k k k per M \operatorname{per} M per M b ∈ [ 2 k ] ∖ [ k ] = { k + 1 , … , 2 k } b \in[2k]\setminus[k]=\{k+1,\ldots,2k\} b ∈ [ 2 k ] ∖ [ k ] = { k + 1 , … , 2 k } k k k per M ‾ \operatorname{per}\overline{M} per M a a a [ k ] [k] [ k ] b b b [ 2 k ] ∖ [ k ] [2k]\setminus[k] [ 2 k ] ∖ [ k ]
∣ per M ∣ 2 k = ∮ d μ 2 k ( u , v ) exp ( ∑ i , j = 1 N [ M i j X i j + + M ‾ i j X i j − ] ) , (40) |\operatorname{per} M|^{2k}=\oint d\mu_{2k}(u,v)
\exp\left(\sum_{i,j=1}^{N}\left[M_{ij}X_{ij}^{+}+\overline{M}_{ij}X_{ij}^{-}\right]\right), \tag*{(40)} ∣ per M ∣ 2 k = ∮ d μ 2 k ( u , v ) exp ( i , j = 1 ∑ N [ M ij X ij + + M ij X ij − ] ) , (40) X i j + = ∑ a = 1 k u i a v j a , X i j − = ∑ b = k + 1 2 k u i b v j b . X_{ij}^{+}=\sum_{a=1}^{k}u_i^a v_j^a,\qquad
X_{ij}^{-}=\sum_{b=k+1}^{2k}u_i^b v_j^b. X ij + = a = 1 ∑ k u i a v j a , X ij − = b = k + 1 ∑ 2 k u i b v j b . so that X i j + + X i j − X_{ij}^{+}+X_{ij}^{-} X ij + + X ij − X i j X_{ij} X ij b > k b>k b > k a ≤ k a\leq k a ≤ k ⟨ e x M i j + y M ‾ i j ⟩ = e x y \langle e^{xM_{ij}+y\overline{M}_{ij}}\rangle=e^{xy} ⟨ e x M ij + y M ij ⟩ = e x y x , y x,y x , y
⟨ ∣ per M ∣ 2 k ⟩ = ∮ d μ 2 k ( u , v ) exp ( ∑ i , j = 1 N X i j + X i j − ) = ∮ d μ 2 k ( u , v ) exp ( ∑ a = 1 k ∑ b = k + 1 2 k R a b S a b ) , (41) \begin{aligned}
\langle|\operatorname{per} M|^{2k}\rangle
=\oint d\mu_{2k}(u,v)\exp\left(\sum_{i,j=1}^{N}X_{ij}^{+}X_{ij}^{-}\right) \\
&=\oint d\mu_{2k}(u,v)\exp\left(\sum_{a=1}^{k}\sum_{b=k+1}^{2k}R_{ab}S_{ab}\right), \tag*{(41)}
\end{aligned} ⟨ ∣ per M ∣ 2 k ⟩ = ∮ d μ 2 k ( u , v ) exp ( i , j = 1 ∑ N X ij + X ij − ) = ∮ d μ 2 k ( u , v ) exp ( a = 1 ∑ k b = k + 1 ∑ 2 k R ab S ab ) , (41) with R a b = ∑ i u i a u i b R_{ab}=\sum_i u_i^a u_i^b R ab = ∑ i u i a u i b S a b = ∑ j v j a v j b S_{ab}=\sum_j v_j^a v_j^b S ab = ∑ j v j a v j b X X X k 2 k^2 k 2 a ∈ [ k ] a \in[k] a ∈ [ k ] b ∈ [ 2 k ] ∖ [ k ] b \in[2k]\setminus[k] b ∈ [ 2 k ] ∖ [ k ] D = k ( 2 k − 1 ) D=k(2k-1) D = k ( 2 k − 1 ) [ 2 k ] [2k] [ 2 k ] ⟨ M i j 2 ⟩ = ⟨ M ‾ i j 2 ⟩ = 0 \langle M_{ij}^2\rangle=\langle\overline{M}_{ij}^{\,2}\rangle=0 ⟨ M ij 2 ⟩ = ⟨ M ij 2 ⟩ = 0 per M \operatorname{per} M per M per M ‾ \operatorname{per}\overline{M} per M K k , k K_{k,k} K k , k [ 2 k ] [2k] [ 2 k ]
Table 2. Moments ⟨ Y k ⟩ = ⟨ ( per M ) 2 k ⟩ / N ! k \langle Y^k\rangle= \langle(\operatorname{per} M)^{2k}\rangle/N!^k ⟨ Y k ⟩ = ⟨( per M ) 2 k ⟩ / N ! k ± 1 \pm1 ± 1 2 49 2^{49} 2 49 N = 8 N=8 N = 8 2 64 2^{64} 2 64 N = 9 N=9 N = 9 e W k R C k R N k / 2 e N Λ R ( k ) e^{W_k^{\mathrm{R}}} C_k^{\mathrm{R}} N^{k/2}e^{N\Lambda_{\mathrm{R}}(k)} e W k R C k R N k /2 e N Λ R ( k ) 1 + ( a 1 R + w 1 ) / N 1+(a_1^{\mathrm{R}}+w_1)/N 1 + ( a 1 R + w 1 ) / N
We decouple each of the k 2 k^2 k 2 κ = 1 \kappa=1 κ = 1 z a b z_{ab} z ab c B = z a b c_B=z_{ab} c B = z ab B = { a , b } B=\{a,b\} B = { a , b } c B = 0 c_B=0 c B = 0 [ 2 k ] [2k] [ 2 k ] a ∈ [ k ] a\in[k] a ∈ [ k ] b ∈ [ 2 k ] ∖ [ k ] b\in[2k]\setminus[k] b ∈ [ 2 k ] ∖ [ k ] σ : [ k ] → [ 2 k ] ∖ [ k ] \sigma:[k]\to[2k]\setminus[k] σ : [ k ] → [ 2 k ] ∖ [ k ]
∑ σ ∏ a = 1 k z a , σ ( a ) = per ( z ) , (42) \sum_{\sigma}\prod_{a=1}^{k}z_{a,\sigma(a)}=\operatorname{per}(z), \tag*{(42)} σ ∑ a = 1 ∏ k z a , σ ( a ) = per ( z ) , (42) the permanent of the k × k k\times k k × k ( z a b ) (z_{ab}) ( z ab ) a ∈ [ k ] a\in[k] a ∈ [ k ] b ∈ [ 2 k ] ∖ [ k ] b\in[2k]\setminus[k] b ∈ [ 2 k ] ∖ [ k ] per ( z ) \operatorname{per}(z) per ( z ) z a a ′ = z b b ′ = 0 z_{aa'}=z_{bb'}=0 z a a ′ = z b b ′ = 0 per ( z ‾ ) = per ( z ) ‾ \operatorname{per}(\overline{z})=\overline{\operatorname{per}(z)} per ( z ) = per ( z )
⟨ ∣ per M ∣ 2 k ⟩ = E z ∣ per ( z ) ∣ 2 N , (43) \langle|\operatorname{per} M|^{2k}\rangle=\mathbb{E}_z|\operatorname{per}(z)|^{2N}, \tag*{(43)} ⟨ ∣ per M ∣ 2 k ⟩ = E z ∣ per ( z ) ∣ 2 N , (43) All the integrals are absolutely convergent, and the contours are compact, so (43) is an exact identity. Since E z \mathbb{E}_z E z k × k k\times k k × k
⟨ ∣ per M ∣ 2 k ⟩ N = ⟨ ∣ per M ∣ 2 N ⟩ k , (44) \langle|\operatorname{per} M|^{2k}\rangle_N=\langle|\operatorname{per} M|^{2N}\rangle_k, \tag*{(44)} ⟨ ∣ per M ∣ 2 k ⟩ N = ⟨ ∣ per M ∣ 2 N ⟩ k , (44) where the subscript denotes the size of M M M
This moment-size duality can be seen as a special case of Theorem 6.2 in [28 ] . [40 ] was obtained in [41 ] using representation theory. Notice that such a duality exists also for the SK model [38 ] in which the number of replicas can be exchanged with the number of spins in the partition function.
The real matrix case instead has no such symmetry and the dual object is a Hafnian. For k = 1 k=1 k = 1 E z ∣ z ∣ 2 N = N ! \mathbb{E}_z|z|^{2N}=N! E z ∣ z ∣ 2 N = N ! k = 2 k=2 k = 2 per ( z ) = z 13 z 24 + z 14 z 23 \operatorname{per}(z)=z_{13}z_{24}+z_{14}z_{23} per ( z ) = z 13 z 24 + z 14 z 23
⟨ ∣ per M ∣ 4 ⟩ = ∑ n = 0 N ( N n ) 2 ( n ! ) 2 ( ( N − n ) ! ) 2 = ( N + 1 ) N ! 2 , (45) \left\langle|\operatorname{per} M|^4\right\rangle
=\sum_{n=0}^{N}\binom{N}{n}^{2}(n!)^{2}((N-n)!)^{2}
=(N+1)N!^{2}, \tag*{(45)} ⟨ ∣ per M ∣ 4 ⟩ = n = 0 ∑ N ( n N ) 2 ( n ! ) 2 (( N − n )! ) 2 = ( N + 1 ) N ! 2 , (45) i.e. ⟨ Y 2 ⟩ = N + 1 \langle Y^2\rangle=N+1 ⟨ Y 2 ⟩ = N + 1 N N N
Rescaling z a b = N y a b z_{ab}=\sqrt{N}\,y_{ab} z ab = N y ab
⟨ ∣ per M ∣ 2 k ⟩ = N k N + k 2 ∫ ∏ a = 1 k ∏ b = k + 1 2 k d 2 y a b π × exp [ − N Φ C ( y ) ] , (46) \left\langle|\operatorname{per} M|^{2k}\right\rangle
=N^{kN+k^2}\int\prod_{a=1}^{k}\prod_{b=k+1}^{2k}\frac{d^2y_{ab}}{\pi}
\times\exp[-N\Phi_{\mathrm{C}}(y)], \tag*{(46)} ⟨ ∣ per M ∣ 2 k ⟩ = N k N + k 2 ∫ a = 1 ∏ k b = k + 1 ∏ 2 k π d 2 y ab × exp [ − N Φ C ( y )] , (46) where Φ C = ∑ a , b ∣ y a b ∣ 2 − ln per ( y ) − ln per ( y ‾ ) \Phi_{\mathrm{C}}=\sum_{a,b}|y_{ab}|^2-\ln\operatorname{per}(y)-\ln\operatorname{per}(\overline{y}) Φ C = ∑ a , b ∣ y ab ∣ 2 − ln per ( y ) − ln per ( y ) k k k 2 k 2 2k^2 2 k 2 N N N ∂ y a b Φ C = 0 \partial_{y_{ab}}\Phi_{\mathrm{C}}=0 ∂ y ab Φ C = 0
y a b = ( ∂ a b per ( y ) per ( y ) ) , (47) y_{ab}=\left(\frac{\partial_{ab}\operatorname{per}(y)}{\operatorname{per}(y)}\right), \tag*{(47)} y ab = ( per ( y ) ∂ ab per ( y ) ) , (47) and homogeneity gives again ∑ a , b ∣ y a b ∣ 2 = k \sum_{a,b}|y_{ab}|^2=k ∑ a , b ∣ y ab ∣ 2 = k y a b = y 0 y_{ab}=y_0 y ab = y 0 per = k ! y 0 k \operatorname{per}=k!y_0^k per = k ! y 0 k ∂ a b per = ( k − 1 ) ! y 0 k − 1 \partial_{ab}\operatorname{per}=(k-1)!y_0^{k-1} ∂ ab per = ( k − 1 )! y 0 k − 1 y 0 = 1 / ( k y 0 ) y_0=1/(ky_0) y 0 = 1/ ( k y 0 )
y 0 2 − 1 k = q 0 C . (48) y_0^2-\frac{1}{k}=q_0^{\mathrm{C}}. \tag*{(48)} y 0 2 − k 1 = q 0 C . (48) Row by row, each replica a ∈ [ k ] a\in[k] a ∈ [ k ] k k k b ∈ [ 2 k ] ∖ [ k ] b\in[2k]\setminus[k] b ∈ [ 2 k ] ∖ [ k ] 2 k − 1 2k-1 2 k − 1 N ! k N!^k N ! k
Λ C ( k ) = k − Φ C ( y s ) = 2 ln Γ ( k + 1 ) − k ln k . (49) \Lambda_{\mathrm{C}}(k)=k-\Phi_{\mathrm{C}}(y_s)=2\ln\Gamma(k+1)-k\ln k. \tag*{(49)} Λ C ( k ) = k − Φ C ( y s ) = 2 ln Γ ( k + 1 ) − k ln k . (49) Both Λ R \Lambda^{\mathrm{R}} Λ R Λ C \Lambda^{\mathrm{C}} Λ C ( 2 k − 1 ) ! ! (2k-1)!! ( 2 k − 1 )!! [ 2 k ] [2k] [ 2 k ] k ! k! k ! k k k K k , k K_{k,k} K k , k [41 ] .
One has Λ C ( 1 ) = Λ C ( 2 ) = 0 \Lambda_{\mathrm{C}}(1)=\Lambda_{\mathrm{C}}(2)=0 Λ C ( 1 ) = Λ C ( 2 ) = 0 Λ C ( 3 ) = ln 4 3 ≃ 0.288 \Lambda_{\mathrm{C}}(3)=\ln\frac{4}{3}\simeq0.288 Λ C ( 3 ) = ln 3 4 ≃ 0.288 Λ R ( 3 ) = ln 9 5 ≃ 0.588 \Lambda_{\mathrm{R}}(3)=\ln\frac{9}{5}\simeq0.588 Λ R ( 3 ) = ln 5 9 ≃ 0.588 ∣ per M ∣ 2 |\operatorname{per} M|^2 ∣ per M ∣ 2
Integrating the Gaussian fluctuations around the saddle point and over its flat directions, and including the next order of the saddle-point expansion (see the Appendix), we find
⟨ Y k ⟩ = C k C ( 1 + a 1 C N + O ( 1 N 2 ) ) N ( k − 1 ) / 2 e N Λ C ( k ) , (50) \langle Y^k\rangle=C_k^{\mathrm{C}}\left(1+\frac{a_1^{\mathrm{C}}}{N}+O\left(\frac{1}{N^2}\right)\right)N^{(k-1)/2}e^{N\Lambda_{\mathrm{C}}(k)}, \tag*{(50)} ⟨ Y k ⟩ = C k C ( 1 + N a 1 C + O ( N 2 1 ) ) N ( k − 1 ) /2 e N Λ C ( k ) , (50) where
C k C = ( 2 π ) k − 1 2 k − 1 2 ( ( k − 1 ) 2 k ( k − 2 ) ) ( k − 1 ) 2 2 , (51) C_k^{\mathrm{C}}=(2\pi)^{\frac{k-1}{2}}k^{-\frac{1}{2}}\left(\frac{(k-1)^2}{k(k-2)}\right)^{\frac{(k-1)^2}{2}}, \tag*{(51)} C k C = ( 2 π ) 2 k − 1 k − 2 1 ( k ( k − 2 ) ( k − 1 ) 2 ) 2 ( k − 1 ) 2 , (51) and
a 1 C ( k ) = ( k − 1 ) ( k 3 − 5 k 2 + 14 k − 12 ) 12 ( k − 2 ) 3 . (52) a_1^{\mathrm{C}}(k)=\frac{(k-1)(k^3-5k^2+14k-12)}{12(k-2)^3}. \tag*{(52)} a 1 C ( k ) = 12 ( k − 2 ) 3 ( k − 1 ) ( k 3 − 5 k 2 + 14 k − 12 ) . (52) The replica-symmetric saddle is the global maximum of ∣ per ( y ) ∣ |\operatorname{per}(y)| ∣ per ( y ) ∣ ∑ a , b ∣ y a b ∣ 2 = k \sum_{a,b}|y_{ab}|^2=k ∑ a , b ∣ y ab ∣ 2 = k k = 3 , 4 , 5 k=3,4,5 k = 3 , 4 , 5 [42 ] . Since for fixed k k k
For k = 2 k=2 k = 2 ( Re ζ a b ) 2 (\operatorname{Re}\zeta_{ab})^2 ( Re ζ ab ) 2 (87) of the Appendix, vanishes, which is why C k C C_k^{\mathrm{C}} C k C a 1 C a_1^{\mathrm{C}} a 1 C ( k − 1 ) 2 = 1 (k-1)^2=1 ( k − 1 ) 2 = 1 N 1 / 2 N^{1/2} N 1/2 ⟨ Y 2 ⟩ = N + 1 \langle Y^2\rangle=N+1 ⟨ Y 2 ⟩ = N + 1 k < 2 k<2 k < 2 Λ C \Lambda_{\mathrm{C}} Λ C k k k k = 0 , 1 , 2 k=0,1,2 k = 0 , 1 , 2 [ 0 , 2 ] [0,2] [ 0 , 2 ] k ≥ 2 k\geq2 k ≥ 2 0 < k < 2 0<k<2 0 < k < 2
In [41 ] the moments are normalized by N N , k = N ! 2 k ! 2 N / ( k N ) ! \mathcal{N}_{N,k}=N!^2k!^{2N}/(kN)! N N , k = N ! 2 k ! 2 N / ( k N )! ⟨ ∣ per M ∣ 2 k ⟩ \langle|\operatorname{per}M|^{2k}\rangle ⟨ ∣ per M ∣ 2 k ⟩ N ↔ k N\leftrightarrow k N ↔ k
( k N ) ! = k k N N ! k k 1 2 ( 2 π N ) − k − 1 2 ( 1 + O ( 1 / N ) ) , (kN)!=k^{kN}N!^k k^{\frac{1}{2}}(2\pi N)^{-\frac{k-1}{2}}\left(1+O(1/N)\right), ( k N )! = k k N N ! k k 2 1 ( 2 π N ) − 2 k − 1 ( 1 + O ( 1/ N ) ) , so that
N N , k = N ! k ( k ! 2 k k ) N ( 2 π N ) k − 1 2 k − 1 2 ( 1 + O ( 1 / N ) ) = N ! k ( 2 π ) k − 1 2 k − 1 2 k k − 1 2 e N Λ C ( k ) ( 1 + O ( 1 / N ) ) , (53) \begin{aligned}
\mathcal{N}_{N,k}=N!^k\left(\frac{k!^2}{k^k}\right)^N(2\pi N)^{\frac{k-1}{2}}k^{-\frac{1}{2}}\left(1+O(1/N)\right) \\
&=N!^k(2\pi)^{\frac{k-1}{2}}k^{-\frac{1}{2}}k^{\frac{k-1}{2}}e^{N\Lambda_{\mathrm{C}}(k)}\left(1+O(1/N)\right), \tag*{(53)}
\end{aligned} N N , k = N ! k ( k k k ! 2 ) N ( 2 π N ) 2 k − 1 k − 2 1 ( 1 + O ( 1/ N ) ) = N ! k ( 2 π ) 2 k − 1 k − 2 1 k 2 k − 1 e N Λ C ( k ) ( 1 + O ( 1/ N ) ) , (53) where we used (49). The normalization N N , k \mathcal{N}_{N,k} N N , k N N N
lim N → ∞ ⟨ ∣ per M ∣ 2 k ⟩ N N , k = ( ( k − 1 ) 2 k ( k − 2 ) ) ( k − 1 ) 2 / 2 , (54) \lim_{N\to\infty}\frac{\langle|\operatorname{per}M|^{2k}\rangle}{\mathcal{N}_{N,k}}=\left(\frac{(k-1)^2}{k(k-2)}\right)^{(k-1)^2/2}, \tag*{(54)} N → ∞ lim N N , k ⟨ ∣ per M ∣ 2 k ⟩ = ( k ( k − 2 ) ( k − 1 ) 2 ) ( k − 1 ) 2 /2 , (54) equal to 16 / 9 16/9 16/9 ( 9 / 8 ) 9 / 2 ≃ 1.699 (9/8)^{9/2}\simeq1.699 ( 9/8 ) 9/2 ≃ 1.699 ( 16 / 15 ) 8 ≃ 1.676 (16/15)^8\simeq1.676 ( 16/15 ) 8 ≃ 1.676 k = 3 , 4 , 5 k=3,4,5 k = 3 , 4 , 5 e ≃ 1.649 \sqrt{e}\simeq1.649 e ≃ 1.649 k → ∞ k\to\infty k → ∞ k × k k\times k k × k [41 ] based on exact moments and a representation-theoretic lower bound equal to 13 / 8 13/8 13/8 k ≥ 4 k\geq4 k ≥ 4 13 / 8 = 1 + 1 2 + 1 8 13/8=1+\frac{1}{2}+\frac{1}{8} 13/8 = 1 + 2 1 + 8 1 [41 ] is the truncation at second order of e = ∑ n ≥ 0 2 − n / n ! \sqrt{e}=\sum_{n\geq0}2^{-n}/n! e = ∑ n ≥ 0 2 − n / n !
In Table III we compare (50) with exact moments. These are obtained by evaluating (43) as a finite sum: expanding per ( z ) N \operatorname{per}(z)^N per ( z ) N E z ∣ ∏ a , b z a b A a b ∣ 2 = ∏ a , b A a b ! \mathbb{E}_{z}\left|\prod_{a,b}z_{ab}^{A_{ab}}\right|^2=\prod_{a,b}A_{ab}! E z ∏ a , b z ab A ab 2 = ∏ a , b A ab ! ⟨ ∣ per M ∣ 2 k ⟩ = ∑ A c N ( A ) 2 ∏ a , b A a b ! \langle|\operatorname{per}M|^{2k}\rangle=\sum_A c_N(A)^2\prod_{a,b}A_{ab}! ⟨ ∣ per M ∣ 2 k ⟩ = ∑ A c N ( A ) 2 ∏ a , b A ab ! A A A k × k k\times k k × k N N N c N ( A ) c_N(A) c N ( A ) N N N [ k ] [k] [ k ] A A A k = 4 k=4 k = 4 [41 ] . Including the factor 1 + a 1 C / N 1+a_1^{\mathrm{C}}/N 1 + a 1 C / N k = 3 k=3 k = 3 − 3.3 % -3.3\% − 3.3% − 0.1 % -0.1\% − 0.1% N = 60 N=60 N = 60 k = 4 k=4 k = 4 N = 10 N=10 N = 10 3 % 3\% 3% 1 / N 1/N 1/ N
Table III. Moments ⟨ Y k ⟩ = ⟨ ∣ per M ∣ 2 k ⟩ / N ! k \langle Y^k\rangle=\langle|\operatorname{per} M|^{2k}\rangle/N!^k ⟨ Y k ⟩ = ⟨ ∣ per M ∣ 2 k ⟩ / N ! k C k e N ( k − 1 ) / 2 e N Λ C ( k ) C^{e}_{k}N^{(k-1)/2}e^{N\Lambda_C(k)} C k e N ( k − 1 ) /2 e N Λ C ( k ) 1 + a 1 c / N 1+a_1^c/N 1 + a 1 c / N
Finally, for a general circularly symmetric law, P ( M ) = P ( e i θ M ) P(M)=P(e^{i\theta}M) P ( M ) = P ( e i θ M ) x y xy x y ln ⟨ e x M + y M ‾ ⟩ = ∑ m ≥ 1 κ m C ( x y ) m / ( m ! ) 2 \ln\langle e^{xM+y\overline{M}}\rangle=\sum_{m\geq1}\kappa_m^{\mathrm{C}}(xy)^m/(m!)^2 ln ⟨ e x M + y M ⟩ = ∑ m ≥ 1 κ m C ( x y ) m / ( m ! ) 2 κ 1 C = ⟨ ∣ M ∣ 2 ⟩ = 1 \kappa_1^{\mathrm{C}}=\langle|M|^2\rangle=1 κ 1 C = ⟨ ∣ M ∣ 2 ⟩ = 1 κ 2 C = − 1 \kappa_2^{\mathrm{C}}=-1 κ 2 C = − 1 X + X^+ X + X − X^- X − κ m C \kappa_m^{\mathrm{C}} κ m C m m m [ k ] [k] [ k ] m m m [ 2 k ] ∖ [ k ] [2k]\setminus[k] [ 2 k ] ∖ [ k ] [ 2 k ] [2k] [ 2 k ] Λ C ( k ) \Lambda_{\mathrm{C}}(k) Λ C ( k )
Hafnian
In Gaussian boson sampling the single photons at the input are replaced by single-mode squeezed states [25 ] . The probability of detecting one photon in each mode of a set S S S 2 N 2N 2 N p S = ∣ hf ( B S ) ∣ 2 / ∏ j cosh r j p_S=|\operatorname{hf}(B_S)|^2/\prod_j\cosh r_j p S = ∣ hf ( B S ) ∣ 2 / ∏ j cosh r j r j r_j r j B = U diag ( tanh r j ) U T B=U\operatorname{diag}(\tanh r_j)U^T B = U diag ( tanh r j ) U T m × m m\times m m × m B S B_S B S 2 N × 2 N 2N\times2N 2 N × 2 N S S S m m m N N N m B S / tanh r \sqrt{m}B_S/\tanh r m B S / tanh r
We therefore consider in this subsection a 2 N × 2 N 2N\times2N 2 N × 2 N M M M M i j = M j i M_{ij}=M_{ji} M ij = M j i M i j M_{ij} M ij i < j i<j i < j d 2 M i j π e − ∣ M i j ∣ 2 \frac{d^2M_{ij}}{\pi}e^{-|M_{ij}|^2} π d 2 M ij e − ∣ M ij ∣ 2
hf ( M ) = ∑ T ∈ T 2 N ∏ { i , j } ∈ T M i j , (55) \operatorname{hf}(M)=\sum_{T\in\mathcal{T}_{2N}}\prod_{\{i,j\}\in T}M_{ij}, \tag*{(55)} hf ( M ) = T ∈ T 2 N ∑ { i , j } ∈ T ∏ M ij , (55) where T 2 N \mathcal{T}_{2N} T 2 N ( 2 N − 1 ) ! ! (2N-1)!! ( 2 N − 1 )!! [ 2 N ] [2N] [ 2 N ] ⟨ ∣ hf M ∣ 2 ⟩ = ( 2 N − 1 ) ! ! \langle|\operatorname{hf}M|^2\rangle=(2N-1)!! ⟨ ∣ hf M ∣ 2 ⟩ = ( 2 N − 1 )!!
⟨ Y k ⟩ = ⟨ ∣ hf M ∣ 2 k ⟩ ( ( 2 N − 1 ) ! ! ) k . (56) \langle Y^k\rangle=\frac{\langle|\operatorname{hf}M|^{2k}\rangle}{((2N-1)!!)^{k}}. \tag*{(56)} ⟨ Y k ⟩ = (( 2 N − 1 )!! ) k ⟨ ∣ hf M ∣ 2 k ⟩ . (56) so that Y Y Y 2 N 2N 2 N N N N
Using the representation (7) to the power 2 k 2k 2 k 2 k × 2 N 2k\times2N 2 k × 2 N u i a u_i^a u i a d μ 2 k ( u ) = ∏ a = 1 2 k ∏ i = 1 2 N d u i a / ( 2 π i ( u i a ) 2 ) d\mu_{2k}(u)=\prod_{a=1}^{2k}\prod_{i=1}^{2N}du_i^a/(2\pi i(u_i^a)^2) d μ 2 k ( u ) = ∏ a = 1 2 k ∏ i = 1 2 N d u i a / ( 2 π i ( u i a ) 2 ) a ∈ [ k ] a\in[k] a ∈ [ k ] k k k hf M \operatorname{hf} M hf M b ∈ [ 2 k ] ∖ [ k ] b\in[2k]\setminus[k] b ∈ [ 2 k ] ∖ [ k ] k k k hf M ‾ \operatorname{hf}\overline{M} hf M ⟨ e x M i j + y M i j ⟩ = e x y \langle e^{xM_{ij}+yM_{ij}}\rangle=e^{xy} ⟨ e x M ij + y M ij ⟩ = e x y
⟨ ∣ hf M ∣ 2 k ⟩ = ∮ d μ 2 k ( u ) exp ( ∑ i < j X i j + X i j − ) , (57) \langle|\operatorname{hf} M|^{2k}\rangle=\oint d\mu_{2k}(u)\exp\left(\sum_{i<j}X_{ij}^{+}X_{ij}^{-}\right), \tag*{(57)} ⟨ ∣ hf M ∣ 2 k ⟩ = ∮ d μ 2 k ( u ) exp ( i < j ∑ X ij + X ij − ) , (57) X i j + = ∑ a = 1 k u i a u j a , X i j − = ∑ b = k + 1 2 k u i b u j b . X_{ij}^{+}=\sum_{a=1}^{k}u_i^a u_j^a,\qquad X_{ij}^{-}=\sum_{b=k+1}^{2k}u_i^b u_j^b. X ij + = a = 1 ∑ k u i a u j a , X ij − = b = k + 1 ∑ 2 k u i b u j b . Writing ∑ i < j X i j + X i j − = 1 2 ∑ a , b [ R a b 2 − ∑ i ( u i a u i b ) 2 ] \sum_{i<j}X_{ij}^{+}X_{ij}^{-}=\frac{1}{2}\sum_{a,b}[R_{ab}^{2}-\sum_i(u_i^a u_i^b)^2] ∑ i < j X ij + X ij − = 2 1 ∑ a , b [ R ab 2 − ∑ i ( u i a u i b ) 2 ]
⟨ ∣ hf M ∣ 2 k ⟩ = ∮ d μ 2 k ( u ) exp ( 1 2 ∑ a = 1 k ∑ b = k + 1 2 k R a b 2 ) , (58) \langle|\operatorname{hf} M|^{2k}\rangle=\oint d\mu_{2k}(u)\exp\left(\frac{1}{2}\sum_{a=1}^{k}\sum_{b=k+1}^{2k}R_{ab}^{2}\right), \tag*{(58)} ⟨ ∣ hf M ∣ 2 k ⟩ = ∮ d μ 2 k ( u ) exp ( 2 1 a = 1 ∑ k b = k + 1 ∑ 2 k R ab 2 ) , (58) R a b = ∑ i = 1 2 N u i a u i b . R_{ab}=\sum_{i=1}^{2N}u_i^a u_i^b. R ab = i = 1 ∑ 2 N u i a u i b . Compared with (41), the product R a b S a b R_{ab}S_{ab} R ab S ab R a b 2 R_{ab}^{2} R ab 2
We decouple (58) with a real Hubbard–Stratonovich field. Defining E t ( ⋅ ) = ∫ ∏ a , b d t a b / 2 π e − ∑ a , b t a b 2 / 2 ( ⋅ ) E_t(\cdot)=\int\prod_{a,b}dt_{ab}/\sqrt{2\pi}\,e^{-\sum_{a,b}t_{ab}^{2}/2}(\cdot) E t ( ⋅ ) = ∫ ∏ a , b d t ab / 2 π e − ∑ a , b t ab 2 /2 ( ⋅ ) E t e ∑ a , b t a b R a b = e 1 2 ∑ a , b R a b 2 E_t e^{\sum_{a,b}t_{ab}R_{ab}}=e^{\frac{1}{2}\sum_{a,b}R_{ab}^{2}} E t e ∑ a , b t ab R ab = e 2 1 ∑ a , b R ab 2 N → k N\to k N → k M → t M\to t M → t
⟨ ∣ hf M ∣ 2 k ⟩ = E t [ ( per t ) 2 N ] , (59) \langle|\operatorname{hf} M|^{2k}\rangle=E_t\left[(\operatorname{per}t)^{2N}\right], \tag*{(59)} ⟨ ∣ hf M ∣ 2 k ⟩ = E t [ ( per t ) 2 N ] , (59) where t t t k × k k\times k k × k 2 N 2N 2 N k × k k\times k k × k N N N k k k E z E_z E z 2 k × 2 k 2k\times2k 2 k × 2 k
The moments of the permanents of real Gaussian matrices and the moments of the Hafnians of symmetric complex Gaussian matrices are thus the same function of two variables, ⟨ ( per M ) 2 N ⟩ N = ⟨ ∣ hf M ∣ 2 N ⟩ 2 k \langle(\operatorname{per} M)^{2N}\rangle_N=\langle|\operatorname{hf} M|^{2N}\rangle_{2k} ⟨( per M ) 2 N ⟩ N = ⟨ ∣ hf M ∣ 2 N ⟩ 2 k M M M [43 , 44 , 45 ] , where both moments and benchmarking were considered.
The real case of the previous section is the limit of large matrices at fixed moment of this function; Gaussian boson sampling requires the opposite limit, the large moments of the permanent of a real matrix of fixed size k k k ⟨ ∣ hf M ∣ 6 ⟩ = E t ( per t ) 8 = 40279680 \langle|\operatorname{hf} M|^6\rangle=E_t(\operatorname{per}t)^8=40279680 ⟨ ∣ hf M ∣ 6 ⟩ = E t ( per t ) 8 = 40279680 2 N = 8 2N=8 2 N = 8 k = 3 k=3 k = 3 k = 1 k=1 k = 1 E t 2 N = ( 2 N − 1 ) ! ! E_t^{2N}=(2N-1)!! E t 2 N = ( 2 N − 1 )!! k = 2 k=2 k = 2 per t = t 11 t 22 + t 12 t 21 \operatorname{per}t=t_{11}t_{22}+t_{12}t_{21} per t = t 11 t 22 + t 12 t 21
⟨ Y 2 ⟩ = ∑ l = 0 N ( N l ) 2 ( 2 N 2 l ) − 1 = 4 N ( 2 N N ) ≃ π N . (60) \langle Y^2\rangle=\sum_{l=0}^{N}\binom{N}{l}^{2}\binom{2N}{2l}^{-1}=\frac{4^N}{\binom{2N}{N}}\simeq\sqrt{\pi N}. \tag*{(60)} ⟨ Y 2 ⟩ = l = 0 ∑ N ( l N ) 2 ( 2 l 2 N ) − 1 = ( N 2 N ) 4 N ≃ π N . (60) For general k k k t a b = N s a b t_{ab}=\sqrt{N}s_{ab} t ab = N s ab
⟨ ∣ hf M ∣ 2 k ⟩ = ( 2 π ) − k 2 / 2 N k N + k 2 / 2 ∫ ∏ a , b d s a b e − N Φ hf ( s ) , \langle|\operatorname{hf} M|^{2k}\rangle=(2\pi)^{-k^2/2}N^{kN+k^2/2}\int\prod_{a,b}ds_{ab}\,e^{-N\Phi_{\operatorname{hf}}(s)}, ⟨ ∣ hf M ∣ 2 k ⟩ = ( 2 π ) − k 2 /2 N k N + k 2 /2 ∫ a , b ∏ d s ab e − N Φ hf ( s ) , Φ hf = 1 2 ∑ a , b s a b 2 − 2 ln ∣ per ( s ) ∣ . (61) \Phi_{\operatorname{hf}}=\frac{1}{2}\sum_{a,b}s_{ab}^{2}-2\ln|\operatorname{per}(s)|. \tag*{(61)} Φ hf = 2 1 a , b ∑ s ab 2 − 2 ln ∣ per ( s ) ∣. (61) The stationarity condition reads s a b = 2 ∂ a b per ( s ) / per ( s ) s_{ab}=2\partial_{ab}\operatorname{per}(s)/\operatorname{per}(s) s ab = 2 ∂ ab per ( s ) / per ( s ) ∑ a , b s a b 2 = 2 k \sum_{a,b}s_{ab}^{2}=2k ∑ a , b s ab 2 = 2 k s a b = s 0 s_{ab}=s_0 s ab = s 0 s 0 2 = 2 / k s_0^2=2/k s 0 2 = 2/ k ( 2 N − 1 ) ! ! k = [ 2 ( 2 N / e ) N ( 1 + O ( 1 / N ) ) ] k (2N-1)!!^k=[\sqrt{2}(2N/e)^N(1+O(1/N))]^k ( 2 N − 1 )! ! k = [ 2 ( 2 N / e ) N ( 1 + O ( 1/ N )) ] k
lim N → ∞ 1 N ln ⟨ Y k ⟩ = 2 ln Γ ( k + 1 ) − k ln k = Λ C ( k ) , (62) \lim_{N\to\infty}\frac{1}{N}\ln\langle Y^k\rangle=2\ln\Gamma(k+1)-k\ln k=\Lambda_C(k), \tag*{(62)} N → ∞ lim N 1 ln ⟨ Y k ⟩ = 2 ln Γ ( k + 1 ) − k ln k = Λ C ( k ) , (62) the same as for the permanents of complex Gaussian matrices, with N N N
The prefactor is different. Since the field is real, the continuous phase orbit of the complex case reduces to the sign changes s a b → ε a η b s a b s_{ab} \to\varepsilon_a\eta_b s_{ab} s ab → ε a η b s ab ε a , η b = ± 1 \varepsilon_a,\eta_b=\pm1 ε a , η b = ± 1 per ( s ) 2 \operatorname{per}(s)^2 per ( s ) 2 2 2 k − 1 2^{2k-1} 2 2 k − 1 s a b = s 0 ( 1 + x a b ) s_{ab}=s_0(1+x_{ab}) s ab = s 0 ( 1 + x ab ) Φ h f \Phi_{\mathrm{hf}} Φ hf Φ C \Phi_{\mathrm{C}} Φ C x a b x_{ab} x ab
⟨ Y k ⟩ = C k h f ( 1 + a 1 h f N + O ( 1 N 2 ) ) e N Λ C ( k ) (63) \langle Y^k\rangle=C_k^{\mathrm{hf}}\left(1+\frac{a_1^{\mathrm{hf}}}{N}+O\left(\frac{1}{N^2}\right)\right)e^{N\Lambda_{\mathrm{C}}(k)} \tag*{(63)} ⟨ Y k ⟩ = C k hf ( 1 + N a 1 hf + O ( N 2 1 ) ) e N Λ C ( k ) (63) with
C k h f = 2 k − 1 2 ( k − 1 k − 2 ) ( k − 1 ) 2 2 (64) C_k^{\mathrm{hf}}=2^{\frac{k-1}{2}}\left(\frac{k-1}{k-2}\right)^{\frac{(k-1)^2}{2}} \tag*{(64)} C k hf = 2 2 k − 1 ( k − 2 k − 1 ) 2 ( k − 1 ) 2 (64) a 1 h f ( k ) = k ( k − 1 ) ( k 2 − 2 k + 2 ) 12 ( k − 2 ) 3 . a_1^{\mathrm{hf}}(k)=\frac{k(k-1)(k^2-2k+2)}{12(k-2)^3}. a 1 hf ( k ) = 12 ( k − 2 ) 3 k ( k − 1 ) ( k 2 − 2 k + 2 ) . The absence of a power of N N N C k h f C_k^{\mathrm{hf}} C k hf a 1 h f a_1^{\mathrm{hf}} a 1 hf k = 2 k=2 k = 2 π N \sqrt{\pi N} π N
In Table IV we compare (63) with exact moments for k = 3 k=3 k = 3 3 × 3 3\times3 3 × 3 E t ( per t ) 2 N \mathbb{E}_{t}(\operatorname{per}t)^{2N} E t ( per t ) 2 N N N N per t \operatorname{per}t per t ( 2 N − 1 ) ! ! E ∣ C ∣ 2 N (2N-1)!!\,\mathbb{E}|C|^{2N} ( 2 N − 1 )!! E ∣ C ∣ 2 N C C C 2 × 2 2\times2 2 × 2 C C C N ! [ σ − 1 ( 1 − 4 p σ + 4 p 2 σ 2 − 8 q 2 σ 3 ) − 1 / 2 ] N!\left[\sigma^{-1}(1-4p\sigma+4p^2\sigma^2-8q^2\sigma^3)^{-1/2}\right] N ! [ σ − 1 ( 1 − 4 p σ + 4 p 2 σ 2 − 8 q 2 σ 3 ) − 1/2 ] p = ∑ i t 2 i 2 p=\sum_i t_{2i}^2 p = ∑ i t 2 i 2 q = 2 t 21 t 22 t 23 q=2t_{21}t_{22}t_{23} q = 2 t 21 t 22 t 23 1 + a 1 h f / N 1+a_1^{\mathrm{hf}}/N 1 + a 1 hf / N 0.2 % 0.2\% 0.2% N = 100 N=100 N = 100
Table IV. Moments ⟨ Y 3 ⟩ = ⟨ ∣ hf M ∣ 6 ⟩ / ( 2 N − 1 ) ! ! 3 \langle Y^3\rangle=\langle|\operatorname{hf} M|^6\rangle/(2N-1)!!^3 ⟨ Y 3 ⟩ = ⟨ ∣ hf M ∣ 6 ⟩ / ( 2 N − 1 )! ! 3 2 N 2N 2 N C 3 h f e N Λ C ( 3 ) = 8 ( 4 / 3 ) N C_3^{\mathrm{hf}} e^{N\Lambda_C(3)}=8(4/3)^N C 3 hf e N Λ C ( 3 ) = 8 ( 4/3 ) N 1 + a 1 h f / N 1+a_1^{\mathrm{hf}}/N 1 + a 1 hf / N a 1 h f ( 3 ) = 5 / 2 a_1^{\mathrm{hf}}(3)=5/2 a 1 hf ( 3 ) = 5/2
From moments to distributions
With all moments known analytically, one could invert the Mellin/Laplace transform to obtain the distribution P ( Y ) P(Y) P ( Y ) k ≤ 2 k\leq2 k ≤ 2 Λ ( k ) \Lambda(k) Λ ( k ) k > 2 k>2 k > 2 k = 2 k=2 k = 2 Λ ( k ) < 0 \Lambda(k)<0 Λ ( k ) < 0 1 < k < 2 1<k<2 1 < k < 2 lim N → ∞ ln ⟨ Y k ⟩ / N = 0 \lim_{N\to\infty}\ln\langle Y^k\rangle/N=0 lim N → ∞ ln ⟨ Y k ⟩ / N = 0 Y Y Y k > 2 k>2 k > 2
In this section, as a representative example for which it is easier to obtain the exact numerical values, we look at the permanents of the case M i j = ± 1 M_{ij}=\pm1 M ij = ± 1 mutatis mutandis for all the other cases examined in this paper.
Quantitatively, one first writes the inversion Mellin formula:
P ( y ) = ∫ B d k 2 π i ⟨ Y k ⟩ y − k − 1 . (65) P(y)=\int_{\mathcal{B}}\frac{dk}{2\pi i}\langle Y^k\rangle y^{-k-1}. \tag*{(65)} P ( y ) = ∫ B 2 π i d k ⟨ Y k ⟩ y − k − 1 . (65) where Bromwich path B \mathcal{B} B canonical distribution e − β H e^{-\beta H} e − β H microcanonical distribution δ ( E − H ) \delta(E-H) δ ( E − H ) β \beta β k ↔ β k\leftrightarrow\beta k ↔ β − ln y ↔ H -\ln y\leftrightarrow H − ln y ↔ H P ( y ) P(y) P ( y )
For exponentially large y y y η = 1 N ln y = O ( 1 ) \eta=\frac{1}{N}\ln y=O(1) η = N 1 ln y = O ( 1 )
P ( y ) ≃ 1 y 1 2 π i ∫ c − i ∞ c + i ∞ C k N k / 2 e N [ Λ ( k ) − k η ] d k . (66) P(y)\simeq\frac{1}{y}\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}C_kN^{k/2}e^{N[\Lambda(k)-k\eta]}\,dk. \tag*{(66)} P ( y ) ≃ y 1 2 π i 1 ∫ c − i ∞ c + i ∞ C k N k /2 e N [ Λ ( k ) − k η ] d k . (66) The integral can be computed by deforming the contour to pass through the saddle point k ∗ k^* k ∗
P ( y ) ≃ 1 y C k ∗ N k ∗ / 2 e N [ Λ ( k ∗ ) − k ∗ η ] 2 π N Λ ′ ′ ( k ∗ ) . (67) P(y)\simeq\frac{1}{y}\frac{C_{k^*}N^{k^*/2}e^{N[\Lambda(k^*)-k^*\eta]}}{\sqrt{2\pi N\Lambda^{\prime\prime}(k^*)}}. \tag*{(67)} P ( y ) ≃ y 1 2 π N Λ ′′ ( k ∗ ) C k ∗ N k ∗ /2 e N [ Λ ( k ∗ ) − k ∗ η ] . (67) with the saddle point equation:
Λ ′ ( k ∗ ) = η = 1 N ln y , (68) \Lambda'(k^*)=\eta=\frac{1}{N}\ln y, \tag*{(68)} Λ ′ ( k ∗ ) = η = N 1 ln y , (68) as long as k ∗ ≥ 2 k^*\geq2 k ∗ ≥ 2 η ≥ η c = Λ ′ ( 2 ) \eta\geq\eta_c=\Lambda'(2) η ≥ η c = Λ ′ ( 2 ) η c = Λ ′ ( 2 ) = 0.36066 \eta_c=\Lambda'(2)=0.36066 η c = Λ ′ ( 2 ) = 0.36066 R \mathbb{R} R 0.15242 0.15242 0.15242 C \mathbb{C} C
Working at leading exponential order, P ( y ) ≃ e − N I ( η + η c ) P(y)\simeq e^{-N I(\eta+\eta_c)} P ( y ) ≃ e − N I ( η + η c )
I ( η ) = { 2 η , 0 < η ≤ η c , 2 k ψ ( k + 1 2 ) − 2 ln Γ ( k + 1 2 ) + ln π − 2 k 2 2 k − 1 , η > η c , I(\eta)=
\begin{cases}
2\eta, & 0<\eta\leq\eta_c,\\
2k\psi\left(k+\frac{1}{2}\right)-2\ln\Gamma\left(k+\frac{1}{2}\right)+\ln\pi-\frac{2k^2}{2k-1}, & \eta>\eta_c,
\end{cases} I ( η ) = { 2 η , 2 k ψ ( k + 2 1 ) − 2 ln Γ ( k + 2 1 ) + ln π − 2 k − 1 2 k 2 , 0 < η ≤ η c , η > η c , η = 2 ln 2 + 2 ψ ( k + 1 2 ) − ln ( 2 k − 1 ) − 2 k 2 k − 1 . (69) \eta=2\ln2+2\psi\left(k+\frac{1}{2}\right)-\ln(2k-1)-\frac{2k}{2k-1}. \tag*{(69)} η = 2 ln 2 + 2 ψ ( k + 2 1 ) − ln ( 2 k − 1 ) − 2 k − 1 2 k . (69) We identify two regimes: 0 < η < η c 0<\eta<\eta_c 0 < η < η c y = O ( 1 ) y=O(1) y = O ( 1 ) η = O ( 1 / N ) \eta=O(1/N) η = O ( 1/ N ) η > η c \eta>\eta_c η > η c η ≤ η c \eta\leq\eta_c η ≤ η c y < e N η c y<e^{N\eta_c} y < e N η c N I ( η ) = 2 N η = 2 ln y NI(\eta)=2N\eta=2\ln y N I ( η ) = 2 N η = 2 ln y
P ( y ) ∼ 1 y e − N I ( η ) ∼ A N y 3 . (70) P(y)\sim\frac{1}{y}e^{-NI(\eta)}\sim\frac{A_N}{y^3}. \tag*{(70)} P ( y ) ∼ y 1 e − N I ( η ) ∼ y 3 A N . (70) Going back to the random variable p = per M p=\operatorname{per}M p = per M P ( p ) ∼ 1 / p 5 P(p)\sim1/p^5 P ( p ) ∼ 1/ p 5 A N A_N A N ln y \ln y ln y ⟨ Y k ⟩ \langle Y^k\rangle ⟨ Y k ⟩ k < 2 k<2 k < 2 Λ ( k ) = 0 \Lambda(k)=0 Λ ( k ) = 0 C k C_k C k k < 2 k<2 k < 2
For η > η c \eta>\eta_c η > η c η ≪ ln N \eta\ll\ln N η ≪ ln N I ( η ) I(\eta) I ( η )
P ( y ) ∼ e − e N y 1 / N / 2 , (71) P(y)\sim e^{-e^{N}y^{1/N}/2}, \tag*{(71)} P ( y ) ∼ e − e N y 1/ N /2 , (71) a stretched exponential with a very small stretching exponent 1 / N 1/N 1/ N log − log \log-\log log − log ± 1 \pm1 ± 1 Figure 3 .
Figure 3. The full counting of the N = 8 , 9 N = 8, 9 N = 8 , 9 M i j = ± 1 M_{ij} = \pm1 M ij = ± 1 N = 8 N = 8 N = 8 N = 9 N = 9 N = 9 P ( y ) P(y) P ( y ) F ( y ) = ∫ y + ∞ d y ′ P ( y ′ ) F(y) = \int_y^{+\infty} \mathrm{d}y' P(y') F ( y ) = ∫ y + ∞ d y ′ P ( y ′ ) P ∼ y − 3 P \sim y^{-3} P ∼ y − 3 F ∼ y − 2 F \sim y^{-2} F ∼ y − 2
Implications for the experiments
Boson sampling with single photons— In boson sampling [36 ] N N N N N N m m m U U U Figure 4 .
Figure 4. N N N N N N m m m U U U
The probability of a collision-free outcome, one photon in each output mode of a set S S S p S = ∣ per U S ∣ 2 p_S=|\operatorname{per}U_S|^2 p S = ∣ per U S ∣ 2 U S U_S U S N × N N\times N N × N U U U S S S m m m N N N M = m U S M=\sqrt{m}U_S M = m U S
Y S = ∣ per M ∣ 2 N ! ≃ C p S , C = ( m N ) , (72) Y_S=\frac{|\operatorname{per}M|^2}{N!}\simeq C p_S,\qquad C=\binom{m}{N}, \tag*{(72)} Y S = N ! ∣ per M ∣ 2 ≃ C p S , C = ( N m ) , (72) is the probability of the outcome in units of that of the uniform distribution on the C C C m ≫ N 2 m\gg N^2 m ≫ N 2 U U U ( m N ) ⟨ ∣ per U S ∣ 2 ⟩ = ∏ j = 0 N − 1 m − j m + j ≃ e − N 2 / m \binom{m}{N}\langle|\operatorname{per}U_S|^2\rangle=\prod_{j=0}^{N-1}\frac{m-j}{m+j}\simeq e^{-N^2/m} ( N m ) ⟨ ∣ per U S ∣ 2 ⟩ = ∏ j = 0 N − 1 m + j m − j ≃ e − N 2 / m U U U ⟨ Y k ⟩ \langle Y^k\rangle ⟨ Y k ⟩ k k k
We consider the linear cross-entropy score C ∑ S q S p S C\sum_S q_Sp_S C ∑ S q S p S q S q_S q S p S p_S p S [46 ] . Averaged over U U U ⟨ Y 2 ⟩ = N + 1 \langle Y^2\rangle=N+1 ⟨ Y 2 ⟩ = N + 1 1 1 1 ∑ j = 0 N 1 / j ! → e \sum_{j=0}^{N}1/j!\to e ∑ j = 0 N 1/ j ! → e q S = per ( ∣ U S ∣ 2 ) q_S=\operatorname{per}(|U_S|^2) q S = per ( ∣ U S ∣ 2 ) M M M ∣ per M ∣ 2 |\operatorname{per}M|^2 ∣ per M ∣ 2 2 # f i x e d p o i n t s 2^{\#\mathrm{fixed\ points}} 2 # fixed points
For comparison, in random circuit sampling the output probabilities follow the Porter-Thomas distribution, ⟨ Y k ⟩ = k ! \langle Y^k\rangle=k! ⟨ Y k ⟩ = k ! 2 2 2
The higher moments determine the statistical cost of such tests. The score is estimated by the average of Y S Y_S Y S M M M ⟨ Y 3 ⟩ − ⟨ Y 2 ⟩ 2 \langle Y^3\rangle-\langle Y^2\rangle^2 ⟨ Y 3 ⟩ − ⟨ Y 2 ⟩ 2 ε \varepsilon ε
M ≃ ( ⟨ Y 3 ⟩ ⟨ Y 2 ⟩ 2 − 1 ) ε − 2 ≃ [ 32 π 9 3 N ( 4 / 3 ) N ( N + 1 ) 2 ( 1 + 2 N ) − 1 ] ε − 2 , (73) \begin{aligned}
\mathcal{M}\simeq\left(\frac{\langle Y^3\rangle}{\langle Y^2\rangle^2}-1\right)\varepsilon^{-2} \\
&\simeq\left[\frac{32\pi}{9\sqrt{3}}\frac{N(4/3)^N}{(N+1)^2}\left(1+\frac{2}{N}\right)-1\right]\varepsilon^{-2}, \tag*{(73)}
\end{aligned} M ≃ ( ⟨ Y 2 ⟩ 2 ⟨ Y 3 ⟩ − 1 ) ε − 2 ≃ [ 9 3 32 π ( N + 1 ) 2 N ( 4/3 ) N ( 1 + N 2 ) − 1 ] ε − 2 , (73) where we used (45) and (50) with C 3 C = 32 π / ( 9 3 ) C_3^{\mathrm{C}}=32\pi/(9\sqrt{3}) C 3 C = 32 π / ( 9 3 ) Λ C ( 3 ) = ln 4 3 \Lambda_C(3)=\ln\frac{4}{3} Λ C ( 3 ) = ln 3 4 a 1 C ( 3 ) = 2 a_1^{\mathrm{C}}(3)=2 a 1 C ( 3 ) = 2
With the exact values of Table III , the factor in brackets is about 10 2 10^2 1 0 2 N = 20 N=20 N = 20 1.6 × 10 4 1.6\times10^4 1.6 × 1 0 4 N = 40 N=40 N = 40 3.4 × 10 6 3.4\times10^6 3.4 × 1 0 6 N = 60 N=60 N = 60 ⟨ Y k ⟩ \langle Y^k\rangle ⟨ Y k ⟩ e N [ Λ C ( 2 k − 1 ) − 2 Λ C ( k ) ] e^{N[\Lambda_C(2k-1)-2\Lambda_C(k)]} e N [ Λ C ( 2 k − 1 ) − 2 Λ C ( k )] M \mathcal{M} M e N [ 2 Λ C ( 4 ) − 3 Λ C ( 3 ) ] ≃ e 0.76 N e^{N[2\Lambda_C(4)-3\Lambda_C(3)]}\simeq e^{0.76N} e N [ 2 Λ C ( 4 ) − 3 Λ C ( 3 )] ≃ e 0.76 N U U U C C C Λ C ( k ) \Lambda_C(k) Λ C ( k ) 1 N ln Y \frac{1}{N}\ln Y N 1 ln Y
The hardness of approximate boson sampling relies on the anti-concentration of the permanent [36 ] , a statement about the left tail, Pr [ Y < 1 / poly ( N ) ] \Pr[Y<1/\operatorname{poly}(N)] Pr [ Y < 1/ poly ( N )] [ 0 , 2 ] [0,2] [ 0 , 2 ] ⟨ Y 2 ⟩ = N + 1 \langle Y^2\rangle=N+1 ⟨ Y 2 ⟩ = N + 1 1 / N 1/N 1/ N
Gaussian boson sampling— With squeezed states at the input, the number of photons is not fixed, and one conditions on detecting 2 N 2N 2 N Figure 5 .
Figure 5. Identical single-mode squeezed vacuum states are injected into all m m m U U U 2 N 2N 2 N
The output probabilities are Hafnians, and we define Y Y Y n n n n = N n = N n = N n = 2 N n = 2N n = 2 N
as e ( n / 2 ) Λ C ( k ) e^{(n/2)\Lambda_{\mathrm{C}}(k)} e ( n /2 ) Λ C ( k ) e n Λ C ( k ) e^{n\Lambda_{\mathrm{C}}(k)} e n Λ C ( k ) n n n
M ≃ [ 8 ( 4 / 3 ) N π N ( 1 + 5 2 N ) − 1 ] ε − 2 , (74) \mathcal{M} \simeq\left[\frac{8(4/3)^N}{\pi N}\left(1+\frac{5}{2N}\right)-1\right]\varepsilon^{-2}, \tag*{(74)} M ≃ [ π N 8 ( 4/3 ) N ( 1 + 2 N 5 ) − 1 ] ε − 2 , (74) which grows as ( 2 / 3 ) n (2/\sqrt{3})^n ( 2/ 3 ) n ( 4 / 3 ) n (4/3)^n ( 4/3 ) n n = 40 n = 40 n = 40 1.6 × 10 4 1.6 \times10^4 1.6 × 1 0 4
In the calculation, the difference comes from the amplitude structure. For single photons, the amplitude involves both the input and the output modes, the rows and the columns of U S U_S U S R a b S a b R_{ab}S_{ab} R ab S ab N ( k − 1 ) / 2 N^{(k-1)/2} N ( k − 1 ) /2 [47 ] , and extensions to immanants in [48 ] .
For squeezed states, the photons are created in pairs, and the amplitude involves only the output modes; the overlaps appear as R a b 2 R_{ab}^2 R ab 2 m ≫ N 2 m \gg N^2 m ≫ N 2 B B B
Conclusions
We showed that an integral representation of the permanent can compute the normalized k k k ⟨ Y k ⟩ \langle Y^k\rangle ⟨ Y k ⟩ N N N k > 2 k > 2 k > 2 k ≤ 2 k \leq2 k ≤ 2
Using this technology, we could compute the overall prefactors and 1 / N 1/N 1/ N O ( 1 / N ) O(1/N) O ( 1/ N )
⟨ ∣ per M ∣ 2 k ⟩ = N ! k e N Λ F ( k ) N α F ( k ) C k F ( 1 + a 1 F ( k ) N ) , (75) \left\langle\lvert\operatorname{per} M\rvert^{2k}\right\rangle= N!^k e^{N\Lambda_{\mathrm{F}}(k)}N^{\alpha_{\mathrm{F}}(k)}C_k^{\mathrm{F}}\left(1+\frac{a_1^{\mathrm{F}}(k)}{N}\right), \tag*{(75)} ⟨ ∣ per M ∣ 2 k ⟩ = N ! k e N Λ F ( k ) N α F ( k ) C k F ( 1 + N a 1 F ( k ) ) , (75) with F = R \mathrm{F} = \mathbb{R} F = R F = C \mathrm{F} = \mathbb{C} F = C α R ( k ) = k / 2 \alpha_{\mathbb{R}}(k)=k/2 α R ( k ) = k /2 α C ( k ) = ( k − 1 ) / 2 \alpha_{\mathbb{C}}(k)=(k-1)/2 α C ( k ) = ( k − 1 ) /2 C k F C_k^{\mathrm{F}} C k F a 1 F a_1^{\mathrm{F}} a 1 F Λ F \Lambda_{\mathrm{F}} Λ F Λ F \Lambda_{\mathrm{F}} Λ F α F \alpha_{\mathrm{F}} α F M i j M_{ij} M ij e W k e^{W_k} e W k (36) , and κ 4 2 \kappa_4^2 κ 4 2 κ 6 \kappa_6 κ 6 1 / N 1/N 1/ N
For the Hafnians of symmetric complex Gaussian matrices, relevant for Gaussian boson sampling, the same asymptotic expansion holds
⟨ ∣ hf M ∣ 2 k ⟩ = ( 2 N − 1 ) ! ! k e N Λ C ( k ) C k h f ( 1 + a 1 h f N ) , (76) \left\langle\lvert\operatorname{hf} M\rvert^{2k}\right\rangle= (2N-1)!!^k e^{N\Lambda_{\mathbb{C}}(k)}C_k^{\mathrm{hf}}\left(1+\frac{a_1^{\mathrm{hf}}}{N}\right), \tag*{(76)} ⟨ ∣ hf M ∣ 2 k ⟩ = ( 2 N − 1 )! ! k e N Λ C ( k ) C k hf ( 1 + N a 1 hf ) , (76) with the rate Λ C \Lambda_{\mathbb{C}} Λ C N N N C k h f C_k^{\mathrm{hf}} C k hf a 1 h f a_1^{\mathrm{hf}} a 1 hf
Acknowledgments
We acknowledge support from INFN through the project “QUANTUM” and from the Italian funding within the “Budget MUR - Dipartimenti di Eccellenza 2023–2027” - Quantum Sensing and Modelling for One-Health (QuaSimOdO). We acknowledge support from Projects PN-RIC Q-SUD (Q-SUD B99H26000380007) and PN-RIC QUANTAS (SYNERGIA B99H26000410007). PF acknowledges support from the Italian National Group of Mathematical Physics (GNFM-IndAM).
Appendix
Compact representations— The formula
det ( M ) = ∫ ∏ i = 1 N d ψ i d ψ ˉ i exp ( ∑ i , j = 1 N ψ ˉ i M i j ψ j ) , (77) \det(M)=\int\prod_{i=1}^{N}d\psi_i d\bar{\psi}_i\exp\left(\sum_{i,j=1}^{N}\bar{\psi}_iM_{ij}\psi_j\right), \tag*{(77)} det ( M ) = ∫ i = 1 ∏ N d ψ i d ψ ˉ i exp ( i , j = 1 ∑ N ψ ˉ i M ij ψ j ) , (77) has a direct generalization to permanents:
per ( M ) = ∫ ∏ i = 1 N d ϕ i d ϕ ˉ i exp ( ∑ i , j = 1 N ϕ ˉ i M i j ϕ j ) , (78) \operatorname{per}(M)=\int\prod_{i=1}^{N}d\phi_i d\bar{\phi}_i\exp\left(\sum_{i,j=1}^{N}\bar{\phi}_iM_{ij}\phi_j\right), \tag*{(78)} per ( M ) = ∫ i = 1 ∏ N d ϕ i d ϕ ˉ i exp ( i , j = 1 ∑ N ϕ ˉ i M ij ϕ j ) , (78) where the ϕ i \phi_i ϕ i
We notice an passant that one has
pf ( M ) = ∫ ∏ i = 1 N d ψ i exp ( ∑ i , j = 1 N ψ i M i j ψ j ) , (79) \operatorname{pf}(M)=\int\prod_{i=1}^{N}d\psi_i\exp\left(\sum_{i,j=1}^{N}\psi_iM_{ij}\psi_j\right), \tag*{(79)} pf ( M ) = ∫ i = 1 ∏ N d ψ i exp ( i , j = 1 ∑ N ψ i M ij ψ j ) , (79) and
hf ( M ) = ∫ ∏ i = 1 N d ϕ i exp ( ∑ i , j = 1 N ϕ i M i j ϕ j ) . (80) \operatorname{hf}(M)=\int\prod_{i=1}^{N}d\phi_i\exp\left(\sum_{i,j=1}^{N}\phi_iM_{ij}\phi_j\right). \tag*{(80)} hf ( M ) = ∫ i = 1 ∏ N d ϕ i exp ( i , j = 1 ∑ N ϕ i M ij ϕ j ) . (80) So determinants, permanents, Pfaffians, and Hafnians have similar compact representations if we introduce nilpotent variables.
Saddle-point expansion— The Gaussian moments of the real and of the complex case have the common form
N k N + D ∫ ∏ e d 2 y e π e − N Φ ( y ) , (81) N^{kN+D}\int\prod_e\frac{d^2y_e}{\pi}e^{-N\Phi(y)}, \tag*{(81)} N k N + D ∫ e ∏ π d 2 y e e − N Φ ( y ) , (81) Φ ( y ) = ∑ e ∣ y e ∣ 2 − ln Ω ( y ) − ln Ω ( y ) ‾ . \Phi(y)=\sum_e\lvert y_e\rvert^2-\ln\Omega(y)-\ln\overline{\Omega(y)}. Φ ( y ) = e ∑ ∣ y e ∣ 2 − ln Ω ( y ) − ln Ω ( y ) . The fields y e y_e y e e e e Γ \Gamma Γ Γ \Gamma Γ [ 2 k ] [2k] [ 2 k ] e = { a , b } e=\{a,b\} e = { a , b } D = k ( 2 k − 1 ) D=k(2k-1) D = k ( 2 k − 1 ) Ω = hf \Omega=\operatorname{hf} Ω = hf Γ = K k , k \Gamma=K_{k,k} Γ = K k , k e = ( a , b ) e=(a,b) e = ( a , b ) a ∈ [ k ] a\in[k] a ∈ [ k ] b ∈ [ 2 k ] ∖ [ k ] b\in[2k]\setminus[k] b ∈ [ 2 k ] ∖ [ k ] D = k 2 D=k^2 D = k 2 Ω = per \Omega=\operatorname{per} Ω = per Ω ( y ) = ∑ T ∈ T ∏ e ∈ T y e \Omega(y)=\sum_{T\in\mathcal{T}}\prod_{e\in T}y_e Ω ( y ) = ∑ T ∈ T ∏ e ∈ T y e T T T Γ \Gamma Γ y e = y 0 y_e=y_0 y e = y 0 y 0 2 = q 0 y_0^2=q_0 y 0 2 = q 0 Γ \Gamma Γ q 0 q_0 q 0 y e = y 0 ( 1 + x e ) y_e=y_0(1+x_e) y e = y 0 ( 1 + x e )
Ω ( y 0 ( 1 + x ) ) Ω ( y 0 1 ) = ∑ n ≥ 0 ω n m n ( x ) , (82) \frac{\Omega(y_0(1+x))}{\Omega(y_0\mathbf{1})}=\sum_{n\geq0}\omega_n m_n(x), \tag*{(82)} Ω ( y 0 1 ) Ω ( y 0 ( 1 + x )) = n ≥ 0 ∑ ω n m n ( x ) , (82) where m n ( x ) m_n(x) m n ( x ) ∏ e ∈ S x e \prod_{e\in S}x_e ∏ e ∈ S x e S S S n n n Γ \Gamma Γ ω n \omega_n ω n Γ \Gamma Γ ω n = ( 2 k − 2 n − 1 ) ! ! / ( 2 k − 1 ) ! ! \omega_n=(2k-2n-1)!!/(2k-1)!! ω n = ( 2 k − 2 n − 1 )!! / ( 2 k − 1 )!! ω n = ( k − n ) ! / k ! \omega_n=(k-n)!/k! ω n = ( k − n )! / k ! ω 1 = q 0 \omega_1=q_0 ω 1 = q 0
Φ ( y ) − Φ ( y s ) = q 0 ∑ e ∣ x e ∣ 2 − 2 Re [ φ ( x ) − q 0 ∑ e x e ] , \Phi(y)-\Phi(y_s)=q_0\sum_e|x_e|^2-2\operatorname{Re}\left[\varphi(x)-q_0\sum_e x_e\right], Φ ( y ) − Φ ( y s ) = q 0 e ∑ ∣ x e ∣ 2 − 2 Re [ φ ( x ) − q 0 e ∑ x e ] , φ ( x ) = ln ∑ n ≥ 0 ω n m n ( x ) = ∑ n ≥ 1 1 n ! ∑ e 1 , … , e n f e 1 … e n ( n ) x e 1 ⋯ x e n . (83) \varphi(x)=\ln\sum_{n\geq0}\omega_n m_n(x)
=\sum_{n\geq1}\frac{1}{n!}\sum_{e_1,\ldots,e_n}f^{(n)}_{e_1\ldots e_n}x_{e_1}\cdots x_{e_n}. \tag*{(83)} φ ( x ) = ln n ≥ 0 ∑ ω n m n ( x ) = n ≥ 1 ∑ n ! 1 e 1 , … , e n ∑ f e 1 … e n ( n ) x e 1 ⋯ x e n . (83) The coefficients f ( n ) f^{(n)} f ( n ) p e 1 … e n ( n ) p^{(n)}_{e_1\ldots e_n} p e 1 … e n ( n ) ω n \omega_n ω n e 1 , … , e n e_1,\ldots,e_n e 1 , … , e n
f e f ( 2 ) = p e f ( 2 ) − q 0 2 , f^{(2)}_{ef}=p^{(2)}_{ef}-q_0^2, f e f ( 2 ) = p e f ( 2 ) − q 0 2 , f e f g ( 3 ) = p e f g ( 3 ) − q 0 ( p e f ( 2 ) + p e g ( 2 ) + p f g ( 2 ) ) + 2 q 0 3 , f^{(3)}_{efg}=p^{(3)}_{efg}-q_0\left(p^{(2)}_{ef}+p^{(2)}_{eg}+p^{(2)}_{fg}\right)+2q_0^3, f e f g ( 3 ) = p e f g ( 3 ) − q 0 ( p e f ( 2 ) + p e g ( 2 ) + p f g ( 2 ) ) + 2 q 0 3 , f e f g h ( 4 ) = p e f g h ( 4 ) − q 0 ∑ [ 4 ] p ( 3 ) − ∑ [ 3 ] p ( 2 ) p ( 2 ) + 2 q 0 2 ∑ [ 6 ] p ( 2 ) − 6 q 0 4 . (84) f^{(4)}_{efgh}=p^{(4)}_{efgh}-q_0\sum_{[4]}p^{(3)}-\sum_{[3]}p^{(2)}p^{(2)}+2q_0^2\sum_{[6]}p^{(2)}-6q_0^4. \tag*{(84)} f e f g h ( 4 ) = p e f g h ( 4 ) − q 0 [ 4 ] ∑ p ( 3 ) − [ 3 ] ∑ p ( 2 ) p ( 2 ) + 2 q 0 2 [ 6 ] ∑ p ( 2 ) − 6 q 0 4 . (84) where ∑ [ n ] \sum_{[n]} ∑ [ n ] n n n e , f , g , h e,f,g,h e , f , g , h
Hessian and moduli space— Let P 0 P_0 P 0 x e = α + ∑ v ∈ e ξ v x_e=\alpha+\sum_{v\in e}\xi_v x e = α + ∑ v ∈ e ξ v v v v Γ \Gamma Γ P 1 = 1 − P 0 P_1=1-P_0 P 1 = 1 − P 0
( P 0 ) e f = ∣ e ∩ f ∣ − 2 q 0 2 k − 2 (real) , (P_0)_{ef}=\frac{|e\cap f|-2q_0}{2k-2}\qquad\text{(real)}, ( P 0 ) e f = 2 k − 2 ∣ e ∩ f ∣ − 2 q 0 (real) , ( P 0 ) ( a b ) , ( a ′ b ′ ) = 1 k 2 + δ a a ′ − 1 k k + δ b b ′ − 1 k k (complex) , (85) (P_0)_{(ab),(a'b')}=\frac{1}{k^2}+\frac{\delta_{aa'}-\frac{1}{k}}{k}+\frac{\delta_{bb'}-\frac{1}{k}}{k}\qquad\text{(complex)}, \tag*{(85)} ( P 0 ) ( ab ) , ( a ′ b ′ ) = k 2 1 + k δ a a ′ − k 1 + k δ b b ′ − k 1 (complex) , (85) of rank n 0 = 2 k n_0=2k n 0 = 2 k n 0 = 2 k − 1 n_0=2k-1 n 0 = 2 k − 1 K k , k K_{k,k} K k , k ξ v \xi_v ξ v + c +c + c − c -c − c x x x x e = α + ξ a + ξ b + ζ e x_e=\alpha+\xi_a+\xi_b+\zeta_e x e = α + ξ a + ξ b + ζ e e = { a , b } e=\{a,b\} e = { a , b } ∑ a = 1 2 k ξ a = 0 \sum_{a=1}^{2k}\xi_a=0 ∑ a = 1 2 k ξ a = 0 P 0 ζ = 0 P_0\zeta=0 P 0 ζ = 0
Φ ( y ) − Φ ( y s ) = 2 k ( Re α ) 2 + 4 ( k − 1 ) 2 k − 1 ∑ a = 1 2 k ( Re ξ a ) 2 + 2 ( k − 2 ) ( 2 k − 1 ) ( 2 k − 3 ) ∑ e ( Re ζ e ) 2 + 2 ( k − 1 ) ( 2 k − 1 ) ( 2 k − 3 ) ∑ e ( Im ζ e ) 2 + O ( x 3 ) . (86) \Phi(y)-\Phi(y_s)=2k(\operatorname{Re}\alpha)^2+\frac{4(k-1)}{2k-1}\sum_{a=1}^{2k}(\operatorname{Re}\xi_a)^2+\frac{2(k-2)}{(2k-1)(2k-3)}\sum_e(\operatorname{Re}\zeta_e)^2+\frac{2(k-1)}{(2k-1)(2k-3)}\sum_e(\operatorname{Im}\zeta_e)^2+O(x^3). \tag*{(86)} Φ ( y ) − Φ ( y s ) = 2 k ( Re α ) 2 + 2 k − 1 4 ( k − 1 ) a = 1 ∑ 2 k ( Re ξ a ) 2 + ( 2 k − 1 ) ( 2 k − 3 ) 2 ( k − 2 ) e ∑ ( Re ζ e ) 2 + ( 2 k − 1 ) ( 2 k − 3 ) 2 ( k − 1 ) e ∑ ( Im ζ e ) 2 + O ( x 3 ) . (86) For complex entries, writing x a b = α + ξ a + η b + ζ a b x_{ab}=\alpha+\xi_a+\eta_b+\zeta_{ab} x ab = α + ξ a + η b + ζ ab ∑ a = 1 k ξ a = ∑ b = k + 1 2 k η b = 0 \sum_{a=1}^k\xi_a=\sum_{b=k+1}^{2k}\eta_b=0 ∑ a = 1 k ξ a = ∑ b = k + 1 2 k η b = 0 ∑ a = 1 k ζ a b = ∑ b = k + 1 2 k ζ a b = 0 \sum_{a=1}^k\zeta_{ab}=\sum_{b=k+1}^{2k}\zeta_{ab}=0 ∑ a = 1 k ζ ab = ∑ b = k + 1 2 k ζ ab = 0
Φ C ( y ) − Φ C ( y s ) = 2 k ( Re α ) 2 + 2 ∑ a = 1 k ( Re ξ a ) 2 + 2 ∑ b = k + 1 2 k ( Re η b ) 2 + k − 2 k ( k − 1 ) ∑ a , b ( Re ζ a b ) 2 + 1 k − 1 ∑ a , b ( Im ζ a b ) 2 + O ( x 3 ) . (87) \Phi_{\mathrm{C}}(y)-\Phi_{\mathrm{C}}(y_s)=2k(\operatorname{Re}\alpha)^2+2\sum_{a=1}^{k}(\operatorname{Re}\xi_a)^2+2\sum_{b=k+1}^{2k}(\operatorname{Re}\eta_b)^2+\frac{k-2}{k(k-1)}\sum_{a,b}(\operatorname{Re}\zeta_{ab})^2+\frac{1}{k-1}\sum_{a,b}(\operatorname{Im}\zeta_{ab})^2+O(x^3). \tag*{(87)} Φ C ( y ) − Φ C ( y s ) = 2 k ( Re α ) 2 + 2 a = 1 ∑ k ( Re ξ a ) 2 + 2 b = k + 1 ∑ 2 k ( Re η b ) 2 + k ( k − 1 ) k − 2 a , b ∑ ( Re ζ ab ) 2 + k − 1 1 a , b ∑ ( Im ζ ab ) 2 + O ( x 3 ) . (87) In both cases the real part of P 0 x P_0x P 0 x 2 q 0 2q_0 2 q 0 2 q 0 ∑ e ( ( P 0 Re x ) e ) 2 2q_0\sum_e((P_0\operatorname{Re}x)_e)^2 2 q 0 ∑ e (( P 0 Re x ) e ) 2 P 1 x P_1x P 1 x λ P \lambda_P λ P λ Q \lambda_Q λ Q λ P \lambda_P λ P k = 2 k=2 k = 2
The imaginary part of P 0 x P_0x P 0 x n 0 n_0 n 0 y e → e i ∑ v ∈ e θ v y e y_e\to e^{i\sum_{v\in e}\theta_v}y_e y e → e i ∑ v ∈ e θ v y e Ω ( y ) \Omega(y) Ω ( y ) e i ∑ v θ v e^{i\sum_v\theta_v} e i ∑ v θ v ∣ Ω ( y ) ∣ |\Omega(y)| ∣Ω ( y ) ∣ ∑ e ∣ y e ∣ 2 \sum_e|y_e|^2 ∑ e ∣ y e ∣ 2 θ ∈ [ 0 , 2 π ) 2 k \theta\in[0,2\pi)^{2k} θ ∈ [ 0 , 2 π ) 2 k θ a \theta_a θ a π \pi π θ a → θ a + θ \theta_a\to\theta_a+\theta θ a → θ a + θ θ b → θ b + θ \theta_b\to\theta_b+\theta θ b → θ b + θ 2 k − 1 2k-1 2 k − 1 ∑ e ∣ d y e ∣ 2 \sum_e|dy_e|^2 ∑ e ∣ d y e ∣ 2
V 0 = 2 k − 1 ( 2 π ) 2 k ( k − 1 2 k − 1 ) k − 1 2 (real) , V_0=2^{k-1}(2\pi)^{2k}\left(\frac{k-1}{2k-1}\right)^{k-\frac12}\qquad\text{(real)}, V 0 = 2 k − 1 ( 2 π ) 2 k ( 2 k − 1 k − 1 ) k − 2 1 (real) , V 0 C = ( 2 π ) 2 k − 1 k − 1 2 (complex) . (88) V_0^{\mathrm{C}}=(2\pi)^{2k-1}k^{-\frac12}\qquad\text{(complex)}. \tag*{(88)} V 0 C = ( 2 π ) 2 k − 1 k − 2 1 (complex) . (88) In the coordinates y e y_e y e Φ \Phi Φ n 0 n_0 n 0 4 4 4 n 0 n_0 n 0 2 λ P / q 0 2\lambda_P/q_0 2 λ P / q 0 2 λ Q / q 0 2\lambda_Q/q_0 2 λ Q / q 0 D − n 0 D-n_0 D − n 0
det ′ H = 2 8 k ( k − 1 ) ( ( k − 1 ) ( k − 2 ) ( 2 k − 3 ) 2 ) k ( 2 k − 3 ) (real) , (89) \det' H = 2^{8k(k-1)}\left(\frac{(k-1)(k-2)}{(2k-3)^2}\right)^{k(2k-3)}
\quad\text{(real)}, \tag*{(89)} det ′ H = 2 8 k ( k − 1 ) ( ( 2 k − 3 ) 2 ( k − 1 ) ( k − 2 ) ) k ( 2 k − 3 ) (real) , (89) det ′ H C = 2 2 k 2 ( k ( k − 2 ) ( k − 1 ) 2 ) ( k − 1 ) 2 (complex) . \det' H_{\mathrm{C}} = 2^{2k^2}\left(\frac{k(k-2)}{(k-1)^2}\right)^{(k-1)^2}
\quad\text{(complex)}. det ′ H C = 2 2 k 2 ( ( k − 1 ) 2 k ( k − 2 ) ) ( k − 1 ) 2 (complex) . Integrating over the r = 2 D − n 0 r=2D-n_0 r = 2 D − n 0 N ! k N!^k N ! k
⟨ Y k ⟩ ≃ ( 2 π ) r − k 2 π − D V 0 det ′ H N n 0 − k 2 e N [ k − Φ ( y s ) ] , (90) \left\langle Y^k\right\rangle\simeq(2\pi)^{\frac{r-k}{2}}\pi^{-D}\frac{V_0}{\sqrt{\det' H}}N^{\frac{n_0-k}{2}}e^{N[k-\Phi(y_s)]}, \tag*{(90)} ⟨ Y k ⟩ ≃ ( 2 π ) 2 r − k π − D det ′ H V 0 N 2 n 0 − k e N [ k − Φ ( y s )] , (90) which gives (30) with (31), and (50) with (51). The power N ( n 0 − k ) / 2 N^{(n_0-k)/2} N ( n 0 − k ) /2 N k / 2 N^{k/2} N k /2 N ( k − 1 ) / 2 N^{(k-1)/2} N ( k − 1 ) /2 C k R C_k^{\mathbb{R}} C k R C k C C_k^{\mathbb{C}} C k C D − n 0 D-n_0 D − n 0 ζ \zeta ζ
The 1 / N 1/N 1/ N To go to the next order we parametrize the integration variables as y e = y 0 e i ∑ v ∈ e θ v ( 1 + x e ) y_e=y_0e^{i\sum_{v\in e}\theta_v}(1+x_e) y e = y 0 e i ∑ v ∈ e θ v ( 1 + x e ) x x x Im P 0 x = 0 \operatorname{Im}P_0x=0 Im P 0 x = 0 y 0 ( 1 + x ) y_0(1+x) y 0 ( 1 + x ) i ( 1 + x e ) ∑ v ∈ e δ θ v i(1+x_e)\sum_{v\in e}\delta\theta_v i ( 1 + x e ) ∑ v ∈ e δ θ v x = 0 x=0 x = 0
J ( x ) J ( 0 ) = det [ P 0 ( 1 + Re x ) P 0 ] , (91) \frac{\mathcal{J}(x)}{\mathcal{J}(0)}=\det\left[P_0(1+\operatorname{Re}x)P_0\right], \tag*{(91)} J ( 0 ) J ( x ) = det [ P 0 ( 1 + Re x ) P 0 ] , (91) where Re x \operatorname{Re}x Re x P 0 P_0 P 0 x → x / N x\to x/\sqrt{N} x → x / N
e − N [ Φ ( y ) − Φ ( y s ) ] J ( x ) J ( 0 ) = e − Φ 2 ( x ) [ 1 + L − Φ 3 N + J 2 − L Φ 3 − Φ 4 + 1 2 Φ 3 2 N + O ( N − 3 / 2 ) ] , (92) e^{-N[\Phi(y)-\Phi(y_s)]}\frac{\mathcal{J}(x)}{\mathcal{J}(0)}
=e^{-\Phi_2(x)}\left[1+\frac{L-\Phi_3}{\sqrt{N}}+\frac{J_2-L\Phi_3-\Phi_4+\frac{1}{2}\Phi_3^2}{N}+O(N^{-3/2})\right], \tag*{(92)} e − N [ Φ ( y ) − Φ ( y s )] J ( 0 ) J ( x ) = e − Φ 2 ( x ) [ 1 + N L − Φ 3 + N J 2 − L Φ 3 − Φ 4 + 2 1 Φ 3 2 + O ( N − 3/2 ) ] , (92) where Φ 2 \Phi_2 Φ 2 Φ n = − 2 Re φ n \Phi_n=-2\operatorname{Re}\varphi_n Φ n = − 2 Re φ n n ≥ 3 n\geq3 n ≥ 3 φ n \varphi_n φ n n n n
L = ∑ e ( P 0 ) e e Re x e = n 0 Re α , L=\sum_e(\mathcal{P}_0)_{ee}\operatorname{Re}x_e=n_0\operatorname{Re}\alpha, L = e ∑ ( P 0 ) ee Re x e = n 0 Re α , J 2 = 1 2 L 2 − 1 2 ∑ e , f ( P 0 ) e f 2 Re x e Re x f . (93) J_2=\frac{1}{2}L^2-\frac{1}{2}\sum_{e,f}(\mathcal{P}_0)_{ef}^2\operatorname{Re}x_e\operatorname{Re}x_f. \tag*{(93)} J 2 = 2 1 L 2 − 2 1 e , f ∑ ( P 0 ) e f 2 Re x e Re x f . (93) Denoting by ⟨ ⋅ ⟩ 0 \langle\cdot\rangle_0 ⟨ ⋅ ⟩ 0 e − Φ 2 e^{-\Phi_2} e − Φ 2 1 / N 1/N 1/ N ⟨ Y k ⟩ \langle Y^k\rangle ⟨ Y k ⟩ a 1 F a_1^{\mathrm{F}} a 1 F F = R , C \mathrm{F}=\mathbb{R},\mathbb{C} F = R , C
a 1 F = ⟨ J 2 ⟩ 0 − ⟨ L Φ 3 ⟩ 0 − ⟨ Φ 4 ⟩ 0 + 1 2 ⟨ Φ 3 2 ⟩ 0 − k 12 , (94) a_1^{\mathrm{F}}=\langle J_2\rangle_0-\langle L\Phi_3\rangle_0-\langle\Phi_4\rangle_0+\frac{1}{2}\langle\Phi_3^2\rangle_0-\frac{k}{12}, \tag*{(94)} a 1 F = ⟨ J 2 ⟩ 0 − ⟨ L Φ 3 ⟩ 0 − ⟨ Φ 4 ⟩ 0 + 2 1 ⟨ Φ 3 2 ⟩ 0 − 12 k , (94) where − k / 12 -k/12 − k /12 N ! k N!^k N ! k σ P , Q = 1 / ( 2 λ P , Q ) \sigma_{P,Q}=1/(2\lambda_{P,Q}) σ P , Q = 1/ ( 2 λ P , Q )
G e f ≡ ⟨ x e x f ⟩ 0 = 1 4 q 0 ( P 0 ) e f + ( σ P − σ Q ) ( P 1 ) e f , \mathcal{G}_{ef}\equiv\langle x_ex_f\rangle_0=\frac{1}{4q_0}(\mathcal{P}_0)_{ef}+(\sigma_P-\sigma_Q)(\mathcal{P}_1)_{ef}, G e f ≡ ⟨ x e x f ⟩ 0 = 4 q 0 1 ( P 0 ) e f + ( σ P − σ Q ) ( P 1 ) e f , K e f ≡ ⟨ x e x ‾ f ⟩ 0 = 1 4 q 0 ( P 0 ) e f + ( σ P + σ Q ) ( P 1 ) e f . (95) \mathcal{K}_{ef}\equiv\langle x_e\overline{x}_f\rangle_0=\frac{1}{4q_0}(\mathcal{P}_0)_{ef}+(\sigma_P+\sigma_Q)(\mathcal{P}_1)_{ef}. \tag*{(95)} K e f ≡ ⟨ x e x f ⟩ 0 = 4 q 0 1 ( P 0 ) e f + ( σ P + σ Q ) ( P 1 ) e f . (95) Since φ \varphi φ
⟨ J 2 ⟩ 0 = 1 4 ∑ e , f [ ( P 0 ) e e ( P 0 ) f f − ( P 0 ) e f 2 ] ( G + K ) e f , \langle J_2\rangle_0=\frac{1}{4}\sum_{e,f}\left[(\mathcal{P}_0)_{ee}(\mathcal{P}_0)_{ff}-(\mathcal{P}_0)_{ef}^2\right](\mathcal{G}+\mathcal{K})_{ef}, ⟨ J 2 ⟩ 0 = 4 1 e , f ∑ [ ( P 0 ) ee ( P 0 ) f f − ( P 0 ) e f 2 ] ( G + K ) e f , − ⟨ L Φ 3 ⟩ 0 = 1 2 ∑ e , f ( P 0 ) e e ( G + K ) e f v f , -\langle L\Phi_3\rangle_0=\frac{1}{2}\sum_{e,f}(\mathcal{P}_0)_{ee}(\mathcal{G}+\mathcal{K})_{ef}v_f, − ⟨ L Φ 3 ⟩ 0 = 2 1 e , f ∑ ( P 0 ) ee ( G + K ) e f v f , − ⟨ Φ 4 ⟩ 0 = 1 4 ∑ e , f , g , h f e f g h ( 4 ) G e f G g h , -\langle\Phi_4\rangle_0=\frac{1}{4}\sum_{e,f,g,h}f^{(4)}_{efgh}\mathcal{G}_{ef}\mathcal{G}_{gh}, − ⟨ Φ 4 ⟩ 0 = 4 1 e , f , g , h ∑ f e f g h ( 4 ) G e f G g h , 1 2 ⟨ Φ 3 2 ⟩ 0 = 1 4 ∑ e , f v e ( G + K ) e f v f + 1 6 ∑ e , f , g , e ′ , f ′ , g ′ f e f g ( 3 ) f e ′ f ′ g ′ ( 3 ) ( G e e ′ G f f ′ G g g ′ + K e e ′ K f f ′ K g g ′ ) . (96) \frac{1}{2}\langle\Phi_3^2\rangle_0=\frac{1}{4}\sum_{e,f}v_e(\mathcal{G}+\mathcal{K})_{ef}v_f+\frac{1}{6}\sum_{e,f,g,e',f',g'}f^{(3)}_{efg}f^{(3)}_{e'f'g'}\left(\mathcal{G}_{ee'}\mathcal{G}_{ff'}\mathcal{G}_{gg'}+\mathcal{K}_{ee'}\mathcal{K}_{ff'}\mathcal{K}_{gg'}\right). \tag*{(96)} 2 1 ⟨ Φ 3 2 ⟩ 0 = 4 1 e , f ∑ v e ( G + K ) e f v f + 6 1 e , f , g , e ′ , f ′ , g ′ ∑ f e f g ( 3 ) f e ′ f ′ g ′ ( 3 ) ( G e e ′ G f f ′ G g g ′ + K e e ′ K f f ′ K g g ′ ) . (96) with v e = ∑ f , g f e f g ( 3 ) G f g v_e=\sum_{f,g}f^{(3)}_{efg}\mathcal{G}_{fg} v e = ∑ f , g f e f g ( 3 ) G f g 1 2 ⟨ Φ 3 2 ⟩ 0 \frac{1}{2}\langle\Phi_3^2\rangle_0 2 1 ⟨ Φ 3 2 ⟩ 0 Figure 6 . Since p ( n ) p^{(n)} p ( n ) G \mathcal{G} G K \mathcal{K} K k k k (97) for real entries and in Eq. (98) for complex entries,
Figure 6. Diagrams of order 1 / N 1/N 1/ N 1 2 ⟨ Φ 3 2 ⟩ 0 \frac{1}{2}\langle\Phi_3^2\rangle_0 2 1 ⟨ Φ 3 2 ⟩ 0 f ( 3 ) f^{(3)} f ( 3 ) φ \varphi φ φ ˉ \bar{\varphi} φ ˉ G e f = ⟨ x e x f ⟩ 0 G_{ef}=\langle x_e x_f\rangle_0 G e f = ⟨ x e x f ⟩ 0 ⟨ x ˉ e x ˉ f ⟩ 0 \langle\bar{x}_e\bar{x}_f\rangle_0 ⟨ x ˉ e x ˉ f ⟩ 0 G G G K e f = ⟨ x e x ˉ f ⟩ 0 K_{ef}=\langle x_e\bar{x}_f\rangle_0 K e f = ⟨ x e x ˉ f ⟩ 0 1 4 ∑ e , f v e G e f v f \frac{1}{4}\sum_{e,f}v_eG_{ef}v_f 4 1 ∑ e , f v e G e f v f 1 4 ∑ e , f v e K e f v f \frac{1}{4}\sum_{e,f}v_eK_{ef}v_f 4 1 ∑ e , f v e K e f v f v e v_e v e 1 6 ∑ e f g ∑ e ′ f ′ g ′ f e f g ( 3 ) f e ′ f ′ g ′ ( 3 ) G e e ′ G f f ′ G g g ′ \frac{1}{6}\sum_{efg}\sum_{e'f'g'}f^{(3)}_{efg}f^{(3)}_{e'f'g'}G_{ee'}G_{ff'}G_{gg'} 6 1 ∑ e f g ∑ e ′ f ′ g ′ f e f g ( 3 ) f e ′ f ′ g ′ ( 3 ) G e e ′ G f f ′ G g g ′ 1 6 ∑ e f g ∑ e ′ f ′ g ′ f e f g ( 3 ) f e ′ f ′ g ′ ( 3 ) K e e ′ K f f ′ K g g ′ \frac{1}{6}\sum_{efg}\sum_{e'f'g'}f^{(3)}_{efg}f^{(3)}_{e'f'g'}K_{ee'}K_{ff'}K_{gg'} 6 1 ∑ e f g ∑ e ′ f ′ g ′ f e f g ( 3 ) f e ′ f ′ g ′ ( 3 ) K e e ′ K f f ′ K g g ′
⟨ J 2 ⟩ 0 = − k ( 2 k − 1 ) ( 2 k − 3 ) 16 ( k − 1 ) ( k − 2 ) , − ⟨ L Φ 3 ⟩ 0 = k ( 2 k 2 − 8 k + 7 ) 4 ( k − 1 ) ( k − 2 ) , − ⟨ Φ 4 ⟩ 0 = − k ( 10 k 5 − 87 k 4 + 296 k 3 − 491 k 2 + 399 k − 128 ) 32 ( k − 1 ) 3 ( k − 2 ) 2 , d u m b b e l l = k ( 2 k 2 − 8 k + 7 ) 2 32 ( k − 1 ) 2 ( k − 2 ) 2 , t h e t a = k ( 20 k 6 − 68 k 5 − 71 k 4 + 564 k 3 − 781 k 2 + 314 k + 31 ) 96 ( k − 1 ) 3 ( k − 2 ) 2 ( 2 k − 5 ) . (97) \begin{aligned}
\langle J_2\rangle_0&=-\frac{k(2k-1)(2k-3)}{16(k-1)(k-2)},&
-\langle L\Phi_3\rangle_0&=\frac{k(2k^2-8k+7)}{4(k-1)(k-2)},\\
-\langle\Phi_4\rangle_0&=-\frac{k(10k^5-87k^4+296k^3-491k^2+399k-128)}{32(k-1)^3(k-2)^2},\\
\mathrm{dumbbell}&=\frac{k(2k^2-8k+7)^2}{32(k-1)^2(k-2)^2},&
\mathrm{theta}&=\frac{k(20k^6-68k^5-71k^4+564k^3-781k^2+314k+31)}{96(k-1)^3(k-2)^2(2k-5)}. \tag*{(97)}
\end{aligned} ⟨ J 2 ⟩ 0 − ⟨ Φ 4 ⟩ 0 dumbbell = − 16 ( k − 1 ) ( k − 2 ) k ( 2 k − 1 ) ( 2 k − 3 ) , = − 32 ( k − 1 ) 3 ( k − 2 ) 2 k ( 10 k 5 − 87 k 4 + 296 k 3 − 491 k 2 + 399 k − 128 ) , = 32 ( k − 1 ) 2 ( k − 2 ) 2 k ( 2 k 2 − 8 k + 7 ) 2 , − ⟨ L Φ 3 ⟩ 0 theta = 4 ( k − 1 ) ( k − 2 ) k ( 2 k 2 − 8 k + 7 ) , = 96 ( k − 1 ) 3 ( k − 2 ) 2 ( 2 k − 5 ) k ( 20 k 6 − 68 k 5 − 71 k 4 + 564 k 3 − 781 k 2 + 314 k + 31 ) . (97) ⟨ J 2 ⟩ 0 = − ( k − 1 ) 2 4 ( k − 2 ) , − ⟨ L Φ 3 ⟩ 0 = ( 2 k − 1 ) ( 2 k 3 − 9 k 2 + 10 k − 4 ) 8 k 2 ( k − 2 ) , − ⟨ Φ 4 ⟩ 0 = − ( 5 k 2 − 6 k + 4 ) ( 2 k 4 − 14 k 3 + 35 k 2 − 34 k + 12 ) 32 k 3 ( k − 2 ) 2 , d u m b b e l l = ( 2 k 3 − 9 k 2 + 10 k − 4 ) 2 32 k 3 ( k − 2 ) 2 , t h e t a = 5 k 7 + 9 k 6 − 113 k 5 + 206 k 4 + 12 k 3 − 360 k 2 + 336 k − 96 48 k 3 ( k − 2 ) 3 . (98) \begin{aligned}
\langle J_2\rangle_0&=-\frac{(k-1)^2}{4(k-2)},&
-\langle L\Phi_3\rangle_0&=\frac{(2k-1)(2k^3-9k^2+10k-4)}{8k^2(k-2)},\\
-\langle\Phi_4\rangle_0&=-\frac{(5k^2-6k+4)(2k^4-14k^3+35k^2-34k+12)}{32k^3(k-2)^2},\\
\mathrm{dumbbell}&=\frac{(2k^3-9k^2+10k-4)^2}{32k^3(k-2)^2},&
\mathrm{theta}&=\frac{5k^7+9k^6-113k^5+206k^4+12k^3-360k^2+336k-96}{48k^3(k-2)^3}. \tag*{(98)}
\end{aligned} ⟨ J 2 ⟩ 0 − ⟨ Φ 4 ⟩ 0 dumbbell = − 4 ( k − 2 ) ( k − 1 ) 2 , = − 32 k 3 ( k − 2 ) 2 ( 5 k 2 − 6 k + 4 ) ( 2 k 4 − 14 k 3 + 35 k 2 − 34 k + 12 ) , = 32 k 3 ( k − 2 ) 2 ( 2 k 3 − 9 k 2 + 10 k − 4 ) 2 , − ⟨ L Φ 3 ⟩ 0 theta = 8 k 2 ( k − 2 ) ( 2 k − 1 ) ( 2 k 3 − 9 k 2 + 10 k − 4 ) , = 48 k 3 ( k − 2 ) 3 5 k 7 + 9 k 6 − 113 k 5 + 206 k 4 + 12 k 3 − 360 k 2 + 336 k − 96 . (98) and their sums with − k / 12 -k/12 − k /12 2 k − 5 2k-5 2 k − 5 ω 3 = 1 / ( ( 2 k − 1 ) ( 2 k − 3 ) ( 2 k − 5 ) ) \omega_3=1/((2k-1)(2k-3)(2k-5)) ω 3 = 1/ (( 2 k − 1 ) ( 2 k − 3 ) ( 2 k − 5 ))
For large k k k a 1 R a_1^{\mathbb{R}} a 1 R a 1 C a_1^{\mathbb{C}} a 1 C k / 12 k/12 k /12 N ! k N!^k N ! k N N , k \mathcal{N}_{N,k} N N , k 1 + ( k 2 − 1 ) / ( 12 k N ) 1+(k^2-1)/(12kN) 1 + ( k 2 − 1 ) / ( 12 k N )
⟨ ∣ per M ∣ 2 k ⟩ N N , k = ( ( k − 1 ) 2 k ( k − 2 ) ) ( k − 1 ) 2 / 2 [ 1 + 2 ( k − 1 ) 3 3 k ( k − 2 ) 3 1 N + O ( N − 2 ) ] . (99) \frac{\langle|\operatorname{per}M|^{2k}\rangle}{\mathcal{N}_{N,k}}
=\left(\frac{(k-1)^2}{k(k-2)}\right)^{(k-1)^2/2}
\left[1+\frac{2(k-1)^3}{3k(k-2)^3}\frac{1}{N}+O(N^{-2})\right]. \tag*{(99)} N N , k ⟨ ∣ per M ∣ 2 k ⟩ = ( k ( k − 2 ) ( k − 1 ) 2 ) ( k − 1 ) 2 /2 [ 1 + 3 k ( k − 2 ) 3 2 ( k − 1 ) 3 N 1 + O ( N − 2 ) ] . (99) whose 1 / N 1/N 1/ N 2 / 3 2/3 2/3 k → ∞ k\to\infty k → ∞ k k k a 1 C ( 3 ) = 2 a_1^{\mathbb{C}}(3)=2 a 1 C ( 3 ) = 2 N = 100 N=100 N = 100 [41 ] .
Hafnians— For the Hafnians of symmetric complex Gaussian matrices the dual integral (59) has real fields, and with s a b = s 0 ( 1 + x a b ) s_{ab}=s_0(1+x_{ab}) s ab = s 0 ( 1 + x ab ) s 0 2 = 2 / k s_0^2=2/k s 0 2 = 2/ k
Φ h f ( s ) − Φ h f ( s s ) = q 0 ∑ a , b x a b 2 − 2 [ φ ( x ) − q 0 ∑ a , b x a b ] , q 0 = 1 k . (100) \begin{aligned}
\Phi_{\mathrm{hf}}(s)-\Phi_{\mathrm{hf}}(s_s)
&=q_0\sum_{a,b}x_{ab}^2-2\left[\varphi(x)-q_0\sum_{a,b}x_{ab}\right],&
q_0&=\frac{1}{k}. \tag*{(100)}
\end{aligned} Φ hf ( s ) − Φ hf ( s s ) = q 0 a , b ∑ x ab 2 − 2 φ ( x ) − q 0 a , b ∑ x ab , q 0 = k 1 . (100) with the same φ \varphi φ x x x R e f ≡ ⟨ x e x f ⟩ 0 = 1 4 q 0 ( P 0 ) e f + σ ( P 1 ) e f = 1 2 ( G + K ) e f R_{ef}\equiv\langle x_ex_f\rangle_0=\frac{1}{4q_0}(P_0)_{ef}+\sigma(P_1)_{ef}=\frac{1}{2}(G+K)_{ef} R e f ≡ ⟨ x e x f ⟩ 0 = 4 q 0 1 ( P 0 ) e f + σ ( P 1 ) e f = 2 1 ( G + K ) e f s a b s_{ab} s ab 2 2 2 2 k − 1 2k-1 2 k − 1 ( k − 2 ) / ( k − 1 ) (k-2)/(k-1) ( k − 2 ) / ( k − 1 ) ( k − 1 ) 2 (k-1)^2 ( k − 1 ) 2 2 2 k − 1 2^{2k-1} 2 2 k − 1 ( 2 N − 1 ) ! ! k (2N-1)!!^k ( 2 N − 1 )! ! k C k h f C_k^{\mathrm{hf}} C k hf
space there are no Jacobian terms, and the 1 / N 1/N 1/ N Figure 6 (a) and (c):
a 1 h f = 1 4 ∑ e , f , g , h f e f g h ( 4 ) R e f R g h + 1 2 ∑ e , f v e R e f v f + 1 3 ∑ e , f , g , e ′ , f ′ , g ′ f e f g ( 3 ) f e ′ f ′ g ′ ( 3 ) R e e ′ R f f ′ R g g ′ + k 24 . (101) a_1^{\mathrm{hf}}=\frac{1}{4}\sum_{e,f,g,h}f^{(4)}_{efgh}R_{ef}R_{gh}
+\frac{1}{2}\sum_{e,f}v_eR_{ef}v_f
+\frac{1}{3}\sum_{e,f,g,e',f',g'}f^{(3)}_{efg}f^{(3)}_{e'f'g'}R_{ee'}R_{ff'}R_{gg'}
+\frac{k}{24}. \tag*{(101)} a 1 hf = 4 1 e , f , g , h ∑ f e f g h ( 4 ) R e f R g h + 2 1 e , f ∑ v e R e f v f + 3 1 e , f , g , e ′ , f ′ , g ′ ∑ f e f g ( 3 ) f e ′ f ′ g ′ ( 3 ) R e e ′ R f f ′ R g g ′ + 24 k . (101) with v e = ∑ f , g f e f g ( 3 ) R f g v_e=\sum_{f,g}f^{(3)}_{efg}R_{fg} v e = ∑ f , g f e f g ( 3 ) R f g k / 24 k/24 k /24 ( 2 N − 1 ) ! ! k (2N-1)!!^k ( 2 N − 1 )! ! k
− ⟨ Φ 4 ⟩ 0 = − k ( 2 k 2 − 6 k + 7 ) 32 ( k − 2 ) 2 , -\langle\Phi_4\rangle_0=-\frac{k(2k^2-6k+7)}{32(k-2)^2}, − ⟨ Φ 4 ⟩ 0 = − 32 ( k − 2 ) 2 k ( 2 k 2 − 6 k + 7 ) , d u m b b e l l = k 32 ( k − 2 ) 2 , \mathrm{dumbbell}=\frac{k}{32(k-2)^2}, dumbbell = 32 ( k − 2 ) 2 k , t h e t a = k ( 5 k 3 − 15 k 2 + 19 k − 10 ) 48 ( k − 2 ) 3 . (102) \mathrm{theta}=\frac{k(5k^3-15k^2+19k-10)}{48(k-2)^3}. \tag*{(102)} theta = 48 ( k − 2 ) 3 k ( 5 k 3 − 15 k 2 + 19 k − 10 ) . (102) and their sum with k / 24 k/24 k /24 a 1 h f a_1^{\mathrm{hf}} a 1 hf k = 3 k=3 k = 3 a 1 h f = 5 / 2 a_1^{\mathrm{hf}}=5/2 a 1 hf = 5/2 N = 300 N=300 N = 300
Disclaimer— Some calculations in this section have been done using Claude and were not checked by hand (although they were checked by an adversarial LLM, ChatGPT, and found correct). In principle, they are straightforward: one needs to expand Φ ( y ) \Phi(y) Φ ( y ) z z z
The 1 / N 1/N 1/ N (94) . For real and complex permanents there are eight of them: the two Jacobian terms ⟨ L 2 ⟩ 0 \langle L^2\rangle_0 ⟨ L 2 ⟩ 0 ⟨ ∑ e , f ( P 0 ) e f 2 Re x e Re x f ⟩ 0 \langle\sum_{e,f}(P_0)^2_{ef}\operatorname{Re}x_e\operatorname{Re}x_f\rangle_0 ⟨ ∑ e , f ( P 0 ) e f 2 Re x e Re x f ⟩ 0 ⟨ L Φ 3 ⟩ 0 \langle L\Phi_3\rangle_0 ⟨ L Φ 3 ⟩ 0 G G G K K K R \mathcal{R} R L = n 0 Re α L=n_0\operatorname{Re}\alpha L = n 0 Re α v e v_e v e k k k k k k N N N
References 
