Introduction
Random growth models and random matrices provide two of the most celebrated examples of universal phenomena in modern probability theory. Over the past three decades, a sequence of remarkable discoveries has uncovered intimate connections between these seemingly unrelated subjects; see e.g. [7 , 19 , 64 , 65 , 83 , 87 , 94 ] for some early work. These developments have shaped many of the central ideas of integrable probability and have led to a detailed understanding of a broad class of exactly solvable stochastic systems.
A landmark result in this direction is due to Johansson [64 ] , who showed that the passage time in last passage percolation (LPP) with i.i.d. geometric weights has the same distribution as the largest particle in the Meixner orthogonal polynomial ensemble. This correspondence transformed a problem in a stochastic growth model into one of random matrix theory, allowing the powerful asymptotic machinery of orthogonal polynomial ensembles to be applied to the analysis of LPP. As a consequence, it led to explicit limit shapes, Tracy–Widom fluctuations, and a detailed understanding of homogeneous LPP. More broadly, it established random matrix theory as one of the central tools of integrable probability.
Since then, many stochastic growth models, interacting particle systems and related integrable systems have been shown to possess analogous algebraic representations through symmetric functions and algebraic combinatorics, leading to determinantal or Pfaffian point processes amenable to asymptotic analysis; see e.g. [4 , 15 , 16 , 22 , 23 , 44 , 45 , 46 , 75 , 82 , 84 ] , as well as the surveys [18 , 21 , 41 , 88 ] . More recently, these ideas have culminated in universal scaling objects such as the KPZ fixed point and the directed landscape [2 , 3 , 46 , 47 , 75 ] . A common feature underlying these developments is the existence of an exactly solvable algebraic structure that serves as a dual description of the original stochastic model.
Despite these remarkable advances, explicit dualities of this kind remain largely confined to homogeneous systems. In many applications, however, one naturally encounters growth in heterogeneous media, where the local growth mechanism varies across space. Such spatial inhomogeneity can drastically alter both the macroscopic geometry and the fluctuation behaviour of the model, while simultaneously obscuring the algebraic structures responsible for exact solvability. Consequently, even determining the limiting shape often becomes a challenging problem.
Motivated by these challenges, various inhomogeneous versions of directed LPP have been investigated. Integrable examples with parameter-dependent geometric weights were studied in [6 , 56 , 66 ] , while more general inhomogeneous environments, including macroscopically varying and discontinuous media, have been studied through hydrodynamic and variational methods [10 , 33 , 38 , 50 , 51 , 67 , 89 ] . Related developments include [11 , 26 , 70 ] . Nevertheless, beyond a small number of special cases, explicit asymptotic descriptions of both the limit shape and the fluctuations remain remarkably scarce.
From the perspective of the general philosophy of integrable probability discussed above, this naturally leads to the following question.
What is the appropriate dual model for an inhomogeneous LPP?
Answering this question requires considerably more than finding another random matrix analogue. The desired dual model should retain enough algebraic structure to admit an exact finite-dimensional correspondence while remaining sufficiently tractable to allow a complete asymptotic analysis.
The principal result of the present paper is that this role is played by the family of q q q q q q q q q [29 , 30 , 31 , 53 , 54 , 55 , 68 , 85 ] and references therein. Specifically, we prove that the passage time in the inhomogeneous geometric LPP [6 , 56 , 66 ] is related, through an explicit logarithmic transformation, to the smallest particle of a suitably parameterised q q q Proposition 4.1 .
This exact correspondence allows the asymptotic theories of the two models to be developed in parallel. On the LPP side, it yields explicit limit shapes together with a complete description of the passage-time fluctuations. On the random matrix side, it leads to a detailed asymptotic theory of the q q q
The present correspondence naturally extends the classical homogeneous picture. As q → 1 q \to1 q → 1 q q q [64 ] .
The rest of this paper is organised as follows. Section 2 introduces the inhomogeneous geometric LPP model and presents our main results on its law of large numbers and fluctuation behaviour. In Section 3, we formulate the corresponding results for the q q q
Figure 1 provides a schematic overview of the main results and the underlying duality framework of the present paper.
Figure 1. Schematic overview of the duality framework and the corresponding asymptotic results for the inhomogeneous geometric LPP model and the q q q
Last passage percolation with inhomogeneous environment
In this section, we introduce the spatially inhomogeneous LPP models [6 , 56 , 66 ] and state our main results. We begin with the geometric model, followed by its exponential analogue, since the latter arises as an extremal limit of the former.
Geometric LPP in the weak and strong inhomogeneity regimes
To incorporate spatial inhomogeneity, let x i , y j ∈ [ 0 , 1 ) x_i,y_j\in[0,1) x i , y j ∈ [ 0 , 1 ) ( i , j ) (i,j) ( i , j )
ω i j ∼ Geom ( x i y j ) , (1) \omega_{ij}\sim\operatorname{Geom}(x_i y_j), \tag*{(1)} ω ij ∼ Geom ( x i y j ) , (1) whose law is
P ( ω i j = k ) = ( 1 − x i y j ) ( x i y j ) k , k = 0 , 1 , 2 , … . (2) \mathbb{P}(\omega_{ij}=k)=(1-x_i y_j)(x_i y_j)^k,\qquad k=0,1,2,\ldots. \tag*{(2)} P ( ω ij = k ) = ( 1 − x i y j ) ( x i y j ) k , k = 0 , 1 , 2 , … . (2) Throughout this paper, we consider the exponentially varying parameters
x i = α q i − 1 , y j = α q j − 1 , (3) x_i=\alpha q^{i-1},\qquad y_j=\alpha q^{j-1}, \tag*{(3)} x i = α q i − 1 , y j = α q j − 1 , (3) where α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) q ∈ ( 0 , 1 ] q\in(0,1] q ∈ ( 0 , 1 ] q = 1 q=1 q = 1 [64 , 90 ] . We note that the parameter q q q q q q
Consider the square lattice equipped with the random vertex weights { ω i j } \{\omega_{ij}\} { ω ij } Π n , m \Pi_{n,m} Π n , m ( 1 , 1 ) (1,1) ( 1 , 1 ) ( n , m ) (n,m) ( n , m ) ( 1 , 0 ) (1,0) ( 1 , 0 ) ( 0 , 1 ) (0,1) ( 0 , 1 )
G n , m = max π ∈ Π n , m ∑ ( i , j ) ∈ π ω i j . (4) G_{n,m}=\max_{\pi\in\Pi_{n,m}}\sum_{(i,j)\in\pi}\omega_{ij}. \tag*{(4)} G n , m = π ∈ Π n , m max ( i , j ) ∈ π ∑ ω ij . (4) Any path attaining the maximum is called a geodesic. The resulting random growth model is referred to as the spatially inhomogeneous geometric LPP model ; see Figure 2 .
Figure 2. LPP realisation for n = m = 20 n=m=20 n = m = 20 α = 0.8 \alpha=0.8 α = 0.8
As previously mentioned, such a spatially inhomogeneous directed LPP and related integrable growth models have been studied extensively. Integrable models with parameter-dependent weights were introduced in [20 , 56 , 66 ] , while variational and PDE approaches to limit shapes were developed in [33 , 38 , 50 , 51 , 89 ] . Spatial inhomogeneity also gives rise to diverse fluctuation behaviours, including Tracy–Widom, Gaussian, and BBP-type limits in various models [10 , 12 , 20 , 49 , 59 , 70 ] . Our model belongs to this class through a deterministic row–column dependent geometric parameter profile.
In the asymptotic analysis of the inhomogeneous LPP, it is natural to choose the parameter q q q (3) according to
q = e − λ / N , λ ∈ ( 0 , ∞ ) , (5) q=e^{-\lambda/N},\qquad\lambda\in(0,\infty), \tag*{(5)} q = e − λ / N , λ ∈ ( 0 , ∞ ) , (5) where N N N n , m = O ( N ) n,m=O(N) n , m = O ( N ) x i x_i x i y j y_j y j q q q q q q i , j = O ( N ) i,j=O(N) i , j = O ( N ) O ( 1 ) O(1) O ( 1 ) q q q [30 , 31 , 32 , 68 ] .
In the scaling regime (2.5), the parameter q q q α \alpha α α \alpha α Figure 3 . This naturally gives rise to two distinct asymptotic regimes.
Figure 3. Weak and strong inhomogeneity regimes in spatially inhomogeneous LPP. The first row shows sample realisations of the random environment { ω i j } \{\omega_{ij}\} { ω ij } { 1 , … , 100 } 2 \{1,\ldots,100\}^{2} { 1 , … , 100 } 2
Weak inhomogeneity. Here α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) ( n , m ) (n,m) ( n , m )
Strong inhomogeneity. Here α = e − η / N \alpha=e^{-\eta/N} α = e − η / N η ∈ ( 0 , ∞ ) \eta\in(0,\infty) η ∈ ( 0 , ∞ )
See Figure 3 for an overview of these two regimes and the corresponding results.
We now state our first result concerning the weak inhomogeneity regime. For this, we recall that the GUE Tracy–Widom distribution F T W ( s ) F_{\mathrm{TW}}(s) F TW ( s )
F T W ( s ) = det ( I − K A i ) L 2 ( s , ∞ ) = exp ( − ∫ s ∞ ( x − s ) u ( x ) 2 d x ) . (6) F_{\mathrm{TW}}(s)=\det\left(I-K_{\mathrm{Ai}}\right)_{L^2(s,\infty)}=\exp\left(-\int_s^\infty(x-s)u(x)^2\,\mathrm{d}x\right). \tag*{(6)} F TW ( s ) = det ( I − K Ai ) L 2 ( s , ∞ ) = exp ( − ∫ s ∞ ( x − s ) u ( x ) 2 d x ) . (6) where
K A i ( x , y ) = A i ( x ) A i ′ ( y ) − A i ′ ( x ) A i ( y ) x − y (7) K_{\mathrm{Ai}}(x,y)=\frac{\mathrm{Ai}(x)\mathrm{Ai}'(y)-\mathrm{Ai}'(x)\mathrm{Ai}(y)}{x-y} \tag*{(7)} K Ai ( x , y ) = x − y Ai ( x ) Ai ′ ( y ) − Ai ′ ( x ) Ai ( y ) (7) is the Airy kernel, and u u u
We introduce the functions which appear in the limit theorem. Fix γ ≥ 1 \gamma\ge1 γ ≥ 1 α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) λ > 0 \lambda> 0 λ > 0
ω ( γ , α , λ ) : = − 2 λ log ( ( 1 − α 2 e − λ ) ( 1 − α 2 e − γ λ ) − α 2 ( 1 − e − λ ) ( 1 − e − γ λ ) 1 − α 2 e − ( γ + 1 ) λ ) . (8) \omega(\gamma,\alpha,\lambda):=-\frac{2}{\lambda}\log\left(\frac{\sqrt{(1-\alpha^2e^{-\lambda})(1-\alpha^2e^{-\gamma\lambda})}-\sqrt{\alpha^2(1-e^{-\lambda})(1-e^{-\gamma\lambda})}}{1-\alpha^2e^{-(\gamma+1)\lambda}}\right). \tag*{(8)} ω ( γ , α , λ ) := − λ 2 log ( 1 − α 2 e − ( γ + 1 ) λ ( 1 − α 2 e − λ ) ( 1 − α 2 e − γ λ ) − α 2 ( 1 − e − λ ) ( 1 − e − γ λ ) ) . (8) We also write
a ≡ a γ , α , λ : = e − λ ( ( 1 − α 2 e − λ ) ( 1 − α 2 e − γ λ ) − α 2 ( 1 − e − λ ) ( 1 − e − γ λ ) 1 − α 2 e − ( γ + 1 ) λ ) 2 , (9) a\equiv a_{\gamma,\alpha,\lambda}:=e^{-\lambda}\left(\frac{\sqrt{(1-\alpha^2e^{-\lambda})(1-\alpha^2e^{-\gamma\lambda})}-\sqrt{\alpha^2(1-e^{-\lambda})(1-e^{-\gamma\lambda})}}{1-\alpha^2e^{-(\gamma+1)\lambda}}\right)^2, \tag*{(9)} a ≡ a γ , α , λ := e − λ ( 1 − α 2 e − ( γ + 1 ) λ ( 1 − α 2 e − λ ) ( 1 − α 2 e − γ λ ) − α 2 ( 1 − e − λ ) ( 1 − e − γ λ ) ) 2 , (9) b ≡ b γ , α , λ : = e − λ ( ( 1 − α 2 e − λ ) ( 1 − α 2 e − γ λ ) + α 2 ( 1 − e − λ ) ( 1 − e − γ λ ) 1 − α 2 e − ( γ + 1 ) λ ) 2 (10) b\equiv b_{\gamma,\alpha,\lambda}:=e^{-\lambda}\left(\frac{\sqrt{(1-\alpha^2e^{-\lambda})(1-\alpha^2e^{-\gamma\lambda})}+\sqrt{\alpha^2(1-e^{-\lambda})(1-e^{-\gamma\lambda})}}{1-\alpha^2e^{-(\gamma+1)\lambda}}\right)^2 \tag*{(10)} b ≡ b γ , α , λ := e − λ ( 1 − α 2 e − ( γ + 1 ) λ ( 1 − α 2 e − λ ) ( 1 − α 2 e − γ λ ) + α 2 ( 1 − e − λ ) ( 1 − e − γ λ ) ) 2 (10) and define
θ γ , α , λ : = { ( 1 − a ) ( 1 − e − ( γ − 1 ) λ b ) − ( 1 − b ) ( 1 − e − ( γ − 1 ) λ a ) λ a ( 1 − a ) ( 1 − e − ( γ − 1 ) λ a ) ( b − a ) , if Ξ γ , α , λ ≥ 1 , ( 1 − a ) ( 1 − e − ( γ − 1 ) λ b ) + ( 1 − b ) ( 1 − e − ( γ − 1 ) λ a ) λ a ( 1 − a ) ( 1 − e − ( γ − 1 ) λ a ) ( b − a ) , if Ξ γ , α , λ < 1 , (11) \theta_{\gamma,\alpha,\lambda}:=
\begin{cases}
\frac{\sqrt{(1-a)(1-e^{-(\gamma-1)\lambda}b)}-\sqrt{(1-b)(1-e^{-(\gamma-1)\lambda}a)}}{\lambda a\sqrt{(1-a)(1-e^{-(\gamma-1)\lambda}a)}(b-a)}, & \text{if }\Xi_{\gamma,\alpha,\lambda}\ge1,\\[1.2em]
\frac{\sqrt{(1-a)(1-e^{-(\gamma-1)\lambda}b)}+\sqrt{(1-b)(1-e^{-(\gamma-1)\lambda}a)}}{\lambda a\sqrt{(1-a)(1-e^{-(\gamma-1)\lambda}a)}(b-a)}, & \text{if }\Xi_{\gamma,\alpha,\lambda}<1,
\end{cases} \tag*{(11)} θ γ , α , λ := ⎩ ⎨ ⎧ λa ( 1 − a ) ( 1 − e − ( γ − 1 ) λ a ) ( b − a ) ( 1 − a ) ( 1 − e − ( γ − 1 ) λ b ) − ( 1 − b ) ( 1 − e − ( γ − 1 ) λ a ) , λa ( 1 − a ) ( 1 − e − ( γ − 1 ) λ a ) ( b − a ) ( 1 − a ) ( 1 − e − ( γ − 1 ) λ b ) + ( 1 − b ) ( 1 − e − ( γ − 1 ) λ a ) , if Ξ γ , α , λ ≥ 1 , if Ξ γ , α , λ < 1 , (11) where
Ξ γ , α , λ : = e − λ ( 1 + α 2 − α 2 e − γ λ ) . (12) \Xi_{\gamma,\alpha,\lambda}:=e^{-\lambda}(1+\alpha^2-\alpha^2e^{-\gamma\lambda}). \tag*{(12)} Ξ γ , α , λ := e − λ ( 1 + α 2 − α 2 e − γ λ ) . (12) We then set
σ ( γ , α , λ ) : = 1 λ a θ γ , α , λ 2 / 3 . (13) \sigma(\gamma,\alpha,\lambda):=\frac{1}{\lambda a\theta_{\gamma,\alpha,\lambda}^{2/3}}. \tag*{(13)} σ ( γ , α , λ ) := λa θ γ , α , λ 2/3 1 . (13) Let ⌊ x ⌋ \lfloor x\rfloor ⌊ x ⌋ x x x
Theorem 2.1 (Geometric LPP at weak inhomogeneity). Let α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) γ ≥ 1 \gamma\ge1 γ ≥ 1 q = e − λ / N q=e^{-\lambda/N} q = e − λ / N λ ∈ ( 0 , ∞ ) \lambda\in(0,\infty) λ ∈ ( 0 , ∞ ) G n , m G_{n,m} G n , m N → ∞ N\to\infty N → ∞
G ⌊ γ N ⌋ , N N → ω ( γ , α , λ ) (14) \frac{G_{\lfloor\gamma N\rfloor,N}}{N}\to\omega(\gamma,\alpha,\lambda) \tag*{(14)} N G ⌊ γ N ⌋ , N → ω ( γ , α , λ ) (14) in probability, where the shape function ω ( γ , α , λ ) \omega(\gamma,\alpha,\lambda) ω ( γ , α , λ ) s ∈ R s\in\mathbb{R} s ∈ R
lim N → ∞ P [ G ⌊ γ N ⌋ , N − N ω ( γ , α , λ ) σ ( γ , α , λ ) N 1 / 3 ≤ s ] = F T W ( s ) , (15) \lim_{N\to\infty}\mathbb{P}\left[\frac{G_{\lfloor\gamma N\rfloor,N}-N\omega(\gamma,\alpha,\lambda)}{\sigma(\gamma,\alpha,\lambda)N^{1/3}}\le s\right]=F_{\mathrm{TW}}(s), \tag*{(15)} N → ∞ lim P [ σ ( γ , α , λ ) N 1/3 G ⌊ γ N ⌋ , N − N ω ( γ , α , λ ) ≤ s ] = F TW ( s ) , (15) where σ ( γ , α , λ ) \sigma(\gamma,\alpha,\lambda) σ ( γ , α , λ )
See Figure 4 for a numerical illustration of Theorem 2.1.
Figure 4. The histogram was made by computing 20000 samples of G ⌊ γ N ⌋ , N − N ω ( γ , α , λ ) σ ( γ , α , λ ) N 1 / 3 \frac{G_{\lfloor\gamma N\rfloor,N}-N\omega(\gamma,\alpha,\lambda)}{\sigma(\gamma,\alpha,\lambda)N^{1/3}} σ ( γ , α , λ ) N 1/3 G ⌊ γ N ⌋ , N − N ω ( γ , α , λ ) N = 400 N=400 N = 400 F T W ( x ) F_{\mathrm{TW}}(x) F TW ( x )
We emphasise that (2.14) provides one of the very few known explicit shape functions for LPP, especially in a spatially inhomogeneous environment. Furthermore, the fluctuation formula (2.15) establishes the Tracy–Widom fluctuations predicted by KPZ universality; cf. [64 Conjecture 1.9] .
Remark 1 (Monotonicity of the shape function). It is straightforward to verify that the shape function ω ( γ , α , λ ) \omega(\gamma,\alpha,\lambda) ω ( γ , α , λ ) λ \lambda λ α \alpha α
E [ ω i j ] = x i y j 1 − x i y j = 1 1 − α 2 q i + j − 2 − 1. (16) \mathbb{E}[\omega_{ij}]=\frac{x_i y_j}{1-x_i y_j}=\frac{1}{1-\alpha^2q^{i+j-2}}-1. \tag*{(16)} E [ ω ij ] = 1 − x i y j x i y j = 1 − α 2 q i + j − 2 1 − 1. (16) Since q = e − λ / N q=e^{-\lambda/N} q = e − λ / N λ \lambda λ x i y j = α 2 q i + j − 2 x_i y_j=\alpha^2q^{i+j-2} x i y j = α 2 q i + j − 2 λ \lambda λ ω ( γ , α , λ ) \omega(\gamma,\alpha,\lambda) ω ( γ , α , λ ) α \alpha α
Remark 2 (Homogeneous limit). In the homogeneous limit λ ↓ 0 \lambda\downarrow0 λ ↓ 0 q ↑ 1 q\uparrow1 q ↑ 1 (3) become x i = y j = α x_i=y_j=\alpha x i = y j = α
lim λ ↓ 0 ω ( γ , α , λ ) = ( 1 + α γ ) 2 1 − α 2 − 1 , (17) \lim_{\lambda\downarrow0}\omega(\gamma,\alpha,\lambda)=\frac{(1+\alpha\sqrt{\gamma})^2}{1-\alpha^2}-1, \tag*{(17)} λ ↓ 0 lim ω ( γ , α , λ ) = 1 − α 2 ( 1 + α γ ) 2 − 1 , (17) which agrees with [64 Theorem 1.1] . Moreover, a direct expansion of (13) gives
lim λ ↓ 0 σ ( γ , α , λ ) = α 1 / 3 ( 1 + α γ ) ( γ + α ) 2 / 3 γ 1 / 6 ( 1 − α 2 ) . (18) \lim_{\lambda\downarrow0}\sigma(\gamma,\alpha,\lambda)=\frac{\alpha^{1/3}(1+\alpha\sqrt{\gamma})(\sqrt{\gamma}+\alpha)^{2/3}}{\gamma^{1/6}(1-\alpha^2)}. \tag*{(18)} λ ↓ 0 lim σ ( γ , α , λ ) = γ 1/6 ( 1 − α 2 ) α 1/3 ( 1 + α γ ) ( γ + α ) 2/3 . (18) This agrees with the fluctuation scale for homogeneous geometric LPP in [64 Theorem 1.2] .
We now turn to the strong inhomogeneity regime. For this, recall that the Bessel kernel is given by
K ν B e s s e l ( x , y ) : = J ν ( x ) y J ν ′ ( y ) − x J ν ′ ( x ) J ν ( y ) 2 ( x − y ) . (19) K_{\nu}^{\mathrm{Bessel}}(x,y):=\frac{J_{\nu}(\sqrt{x})\sqrt{y}J_{\nu}'(\sqrt{y})-\sqrt{x}J_{\nu}'(\sqrt{x})J_{\nu}(\sqrt{y})}{2(x-y)}. \tag*{(19)} K ν Bessel ( x , y ) := 2 ( x − y ) J ν ( x ) y J ν ′ ( y ) − x J ν ′ ( x ) J ν ( y ) . (19) For the strong inhomogeneity regime, we also define
κ γ , λ : = 2 λ e λ ( 1 − e − λ ) ( 1 − e − γ λ ) . (20) \kappa_{\gamma,\lambda}:=\frac{2}{\lambda}\sqrt{e^{\lambda}(1-e^{-\lambda})(1-e^{-\gamma\lambda})}. \tag*{(20)} κ γ , λ := λ 2 e λ ( 1 − e − λ ) ( 1 − e − γ λ ) . (20) Then we have the following result.
Theorem 2.2 (Geometric LPP at strong inhomogeneity). Let α = e − η / N \alpha=e^{-\eta/N} α = e − η / N η ∈ ( 0 , ∞ ) \eta\in(0,\infty) η ∈ ( 0 , ∞ ) γ ≥ 1 \gamma\ge1 γ ≥ 1 q = e − λ / N q=e^{-\lambda/N} q = e − λ / N λ ∈ ( 0 , ∞ ) \lambda\in(0,\infty) λ ∈ ( 0 , ∞ ) G n , m G_{n,m} G n , m (3) . Then as N → ∞ N\to\infty N → ∞
G ⌊ γ N ⌋ , N N log N → 2 λ (21) \frac{G_{\lfloor\gamma N\rfloor,N}}{N\log N}\to\frac{2}{\lambda} \tag*{(21)} N log N G ⌊ γ N ⌋ , N → λ 2 (21) in probability. Furthermore, for s ∈ R s\in\mathbb{R} s ∈ R
lim N → ∞ P [ G ⌊ γ N ⌋ , N − 2 λ N log N N / λ ≤ s ] = det ( I − K a B e s s e l ) L 2 ( 0 , κ γ , λ 2 e − λ e − s ) (22) \lim_{N\to\infty}\mathbb{P}\left[\frac{G_{\lfloor\gamma N\rfloor,N}-\frac{2}{\lambda}N\log N}{N/\lambda}\le s\right]=\det\left(I-K_{a}^{\mathrm{Bessel}}\right)_{L^2(0,\kappa_{\gamma,\lambda}^2e^{-\lambda}e^{-s})} \tag*{(22)} N → ∞ lim P [ N / λ G ⌊ γ N ⌋ , N − λ 2 N log N ≤ s ] = det ( I − K a Bessel ) L 2 ( 0 , κ γ , λ 2 e − λ e − s ) (22) where κ γ , λ \kappa_{\gamma,\lambda} κ γ , λ (20) and
a = 2 η λ − 1. (23) a=\frac{2\eta}{\lambda}-1. \tag*{(23)} a = λ 2 η − 1. (23) See Figure 5 for a numerical illustration of Theorem 2.2 .
Figure 5. The histogram was made by computing 10000 samples of G ⌊ γ N ⌋ , N − 2 λ N log N N / λ \frac{G_{\lfloor\gamma N\rfloor,N}-\frac{2}{\lambda}N\log N}{N/\lambda} N / λ G ⌊ γ N ⌋ , N − λ 2 N l o g N N = 400 N=400 N = 400
A particularly notable feature of Theorem 2.2 is the emergence of hard-edge Bessel fluctuations. While Tracy–Widom fluctuations are ubiquitous in integrable probability, limiting laws governed by the hard-edge Bessel kernel are comparatively rare. Related Bessel fluctuations on an infinite quadrant were obtained in [14 Theorem 2.21] .
Exponential inhomogeneous LPP
We now consider the spatially inhomogeneous LPP with exponential weights. As in the homogeneous setting [64 ] , the exponential model arises as a scaling limit of the geometric model, which usually leads to a considerably simpler theory.
We now introduce the model and state the corresponding limit theorems. Let Exp ( t ) \operatorname{Exp}(t) Exp ( t ) t > 0 t>0 t > 0
P ( X ∈ d x ) = t e − t x d x , x ≥ 0. (25) \mathbb{P}(X\in dx)=te^{-tx}\,dx,\qquad x\ge0. \tag*{(25)} P ( X ∈ d x ) = t e − t x d x , x ≥ 0. (25) Assign to each lattice site ( i , j ) (i,j) ( i , j )
ω ~ i j ∼ Exp ( t i + t j ) . (26) \widetilde{\omega}_{ij}\sim\operatorname{Exp}(t_i+t_j). \tag*{(26)} ω ij ∼ Exp ( t i + t j ) . (26) For given λ , η > 0 \lambda,\eta>0 λ , η > 0
t k = λ ( k + β ) , β : = η λ − 1. (27) t_k=\lambda(k+\beta),\qquad\beta:=\frac{\eta}{\lambda}-1. \tag*{(27)} t k = λ ( k + β ) , β := λ η − 1. (27) The associated last passage time is defined by
H n , m = max π ∈ Π n , m ∑ ( i , j ) ∈ π ω ~ i j . (28) H_{n,m}=\max_{\pi\in\Pi_{n,m}}\sum_{(i,j)\in\pi}\widetilde{\omega}_{ij}. \tag*{(28)} H n , m = π ∈ Π n , m max ( i , j ) ∈ π ∑ ω ij . (28) We refer to this as the spatially inhomogeneous exponential LPP model. The special case γ = λ = 1 \gamma=\lambda=1 γ = λ = 1 [66 ] ; see also [20 ] . For this model, we have the following.
Theorem 2.3 (LPP with inhomogeneous exponential weight). Let η , λ > 0 \eta,\lambda>0 η , λ > 0 γ ≥ 1 \gamma\geq1 γ ≥ 1 H n , m H_{n,m} H n , m (28) . Then as N → ∞ N\to\infty N → ∞
H ⌊ γ N ⌋ , N log N ⟶ 2 λ (29) \frac{H_{\lfloor\gamma N\rfloor,N}}{\log N}\longrightarrow\frac{2}{\lambda} \tag*{(29)} log N H ⌊ γ N ⌋ , N ⟶ λ 2 (29) in probability. Furthermore, for ξ ∈ R \xi\in\mathbb{R} ξ ∈ R
lim N → ∞ P [ H ⌊ γ N ⌋ , N − 2 λ log N ≤ ξ + log γ λ ] = det ( I − K β ) L 2 ( ξ , ∞ ) , (30) \lim_{N\to\infty}\mathbb{P}\left[H_{\lfloor\gamma N\rfloor,N}-\frac{2}{\lambda}\log N\leq\frac{\xi+\log\gamma}{\lambda}\right]=\det(I-K_\beta)_{L^2(\xi,\infty)}, \tag*{(30)} N → ∞ lim P [ H ⌊ γ N ⌋ , N − λ 2 log N ≤ λ ξ + log γ ] = det ( I − K β ) L 2 ( ξ , ∞ ) , (30) where
K β ( x , y ) = 4 e − x + y 2 K 2 β + 1 B e s s e l ( 4 e − x , 4 e − y ) . (31) K_\beta(x,y)=4e^{-\frac{x+y}{2}}K^{\mathrm{Bessel}}_{2\beta+1}(4e^{-x},4e^{-y}). \tag*{(31)} K β ( x , y ) = 4 e − 2 x + y K 2 β + 1 Bessel ( 4 e − x , 4 e − y ) . (31) Here, K ν B e s s e l K^{\mathrm{Bessel}}_\nu K ν Bessel (19) .
See Figure 6 for a numerical illustration of Theorem 2.3 .
Figure 6. The histogram was made by computing 1000 samples of λ ( H ⌊ γ N ⌋ , N − 2 λ log N ) − log γ \lambda(H_{\lfloor\gamma N\rfloor,N}-\frac{2}{\lambda}\log N)-\log\gamma λ ( H ⌊ γ N ⌋ , N − λ 2 log N ) − log γ N = 400 N=400 N = 400 γ = λ = 1 \gamma=\lambda=1 γ = λ = 1 det ( I − K β ) L 2 ( ξ , ∞ ) \det(I-K_\beta)_{L^2(\xi,\infty)} det ( I − K β ) L 2 ( ξ , ∞ )
As in Theorem 2.2 , the independence of the shape function from the aspect ratio γ \gamma γ (30) interpolates between the Gumbel and Tracy–Widom distributions under the appropriate scaling limits; see [17 , 66 ] .
For the special case γ = λ = 1 \gamma=\lambda=1 γ = λ = 1 Theorem 2.3 already appears in [66 Eq. (1.6)] , where Johansson attributes it to Forrester as a private communication; see also [56 Section 2.5] for related discussions. For the square case and β ≥ − 1 / 2 \beta\geq-1/2 β ≥ − 1/2 [14 Theorem 2.25] after a logarithmic change of variables. Here we give a direct derivation for all γ ≥ 1 \gamma\geq1 γ ≥ 1 β > − 1 \beta>-1 β > − 1
We note that, unlike the geometric case considered in this paper, Theorem 2.3 can also be deduced from earlier results, notably [20 ] (see also [13 ] ), via a suitable duality for the inhomogeneous model; see Remark 8. Nevertheless, as a by-product of our analysis of the geometric model, we obtain a direct and self-contained proof of Theorem 2.3 , which is of independent interest.
q q q This section is devoted to the results for the q q q
In general, the concept of q q q q q q q q q [5 , 55 ] , Macdonald polynomial theory [9 , 42 , 72 , 76 , 85 ] , q q q [34 , 36 , 69 , 74 , 79 , 81 , 92 ] , moment and combinatorial formulas [30 , 39 , 55 , 80 , 96 , 98 ] , stochastic processes and martingale structures [24 , 25 ] , tiling models and Gaussian free field fluctuations [48 , 56 , 58 ] , and connections with supersymmetric gauge theories [1 , 35 , 73 , 91 , 93 ] . We refer the reader to [29 , 30 , 31 ] and the references therein for further background.
The q q q q → 1 q \to1 q → 1 q q q
The first is the classical JUE [52 Chapter 3] . It is a system x = { x j } j = 1 N \boldsymbol{x} = \{x_j\}_{j=1}^{N} x = { x j } j = 1 N 0 < x j < 1 0 < x_j < 1 0 < x j < 1
P N J ( d x ) = 1 Z N J ∏ 1 ≤ j < k ≤ N ( x j − x k ) 2 ∏ j = 1 N x j a ( 1 − x j ) b d x j , (32) \mathbb{P}^{\mathrm{J}}_N(d\boldsymbol{x}) = \frac{1}{Z_N^{\mathrm{J}}} \prod_{1 \le j < k \le N}(x_j-x_k)^2 \prod_{j=1}^{N}x_j^a(1-x_j)^b\,dx_j, \tag*{(32)} P N J ( d x ) = Z N J 1 1 ≤ j < k ≤ N ∏ ( x j − x k ) 2 j = 1 ∏ N x j a ( 1 − x j ) b d x j , (32) where a , b > − 1 a,b>-1 a , b > − 1 Z N J Z_N^{\mathrm{J}} Z N J [57 ] .
The second example is the Meixner orthogonal polynomial ensemble [64 ] . For α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) β > 0 \beta> 0 β > 0 h = { h j } j = 1 N \boldsymbol{h} = \{h_j\}_{j=1}^{N} h = { h j } j = 1 N Z ≥ 0 \mathbb{Z}_{\ge0} Z ≥ 0 Z ≥ 0 \mathbb{Z}_{\ge0} Z ≥ 0
P N M ( h ) = 1 Z N M ∏ 1 ≤ j < k ≤ N ( h j − h k ) 2 ∏ j = 1 N ( β ) h j h j ! α 2 h j , (33) \mathbb{P}^{\mathrm{M}}_N(\boldsymbol{h}) = \frac{1}{Z_N^{\mathrm{M}}} \prod_{1 \le j < k \le N}(h_j-h_k)^2 \prod_{j=1}^{N}\frac{(\beta)_{h_j}}{h_j!}\alpha^{2h_j}, \tag*{(33)} P N M ( h ) = Z N M 1 1 ≤ j < k ≤ N ∏ ( h j − h k ) 2 j = 1 ∏ N h j ! ( β ) h j α 2 h j , (33) where ( β ) h (\beta)_h ( β ) h Z N M Z_N^{\mathrm{M}} Z N M
We now introduce the q q q q q q
( z ; q ) k : = ∏ j = 0 k − 1 ( 1 − z q j ) , ( z ; q ) ∞ : = ∏ j = 0 ∞ ( 1 − z q j ) , (34) (z;q)_k := \prod_{j=0}^{k-1}(1-zq^j), \qquad(z;q)_\infty:= \prod_{j=0}^{\infty}(1-zq^j), \tag*{(34)} ( z ; q ) k := j = 0 ∏ k − 1 ( 1 − z q j ) , ( z ; q ) ∞ := j = 0 ∏ ∞ ( 1 − z q j ) , (34) with the convention ( z ; q ) 0 = 1 (z;q)_0=1 ( z ; q ) 0 = 1 a , b > − 1 a,b>-1 a , b > − 1
w ( a , b ) ( x ; q ) = x a ( q x ; q ) ∞ ( q b + 1 x ; q ) ∞ , (35) w^{(a,b)}(x;q) = x^a\frac{(qx;q)_\infty}{(q^{b+1}x;q)_\infty}, \tag*{(35)} w ( a , b ) ( x ; q ) = x a ( q b + 1 x ; q ) ∞ ( q x ; q ) ∞ , (35) which is the little q q q [55 Section 4.1] . This weight belongs to the Askey–Wilson scheme of orthogonal polynomials and is associated with the little q q q x = { x j } j = 1 N \boldsymbol{x}=\{x_j\}_{j=1}^{N} x = { x j } j = 1 N q q q { 1 , q , q 2 , … } \{1,q,q^2,\ldots\} { 1 , q , q 2 , … } x j = q h j x_j=q^{h_j} x j = q h j h j ∈ Z ≥ 0 h_j\in\mathbb{Z}_{\ge0} h j ∈ Z ≥ 0
d q x = ( 1 − q ) ∑ h = 0 ∞ q h δ q h ( d x ) (36) d_qx=(1-q)\sum_{h=0}^{\infty}q^h\delta_{q^h}(dx) \tag*{(36)} d q x = ( 1 − q ) h = 0 ∑ ∞ q h δ q h ( d x ) (36) denote the Jackson measure on [ 0 , 1 ] [0,1] [ 0 , 1 ] x \boldsymbol{x} x
P N ( d x ) = 1 Z N ∏ 1 ≤ j < k ≤ N ( x j − x k ) 2 ∏ j = 1 N w ( a , b ) ( x j ; q ) d q x j , (37) \mathbb{P}_N(d\boldsymbol{x})=\frac{1}{Z_N}\prod_{1\le j<k\le N}(x_j-x_k)^2\prod_{j=1}^{N}w^{(a,b)}(x_j;q)\,d_qx_j, \tag*{(37)} P N ( d x ) = Z N 1 1 ≤ j < k ≤ N ∏ ( x j − x k ) 2 j = 1 ∏ N w ( a , b ) ( x j ; q ) d q x j , (37) where Z N Z_N Z N x j = q h j x_j=q^{h_j} x j = q h j h = { h j } j = 1 N \boldsymbol{h}=\{h_j\}_{j=1}^{N} h = { h j } j = 1 N Z ≥ 0 \mathbb{Z}_{\ge0} Z ≥ 0 Z ≥ 0 N \mathbb{Z}_{\geq0}^{N} Z ≥ 0 N
P N ( h ) = 1 Z ~ N ∏ 1 ≤ j < k ≤ N ( q h j − q h k ) 2 ∏ j = 1 N ( q b + 1 ; q ) h j ( q ; q ) h j q ( a + 1 ) h j , (38) \mathbb{P}_{N}(\boldsymbol{h}) = \frac{1}{\widetilde{Z}_{N}} \prod_{1\leq j<k\leq N}(q^{h_j}-q^{h_k})^2 \prod_{j=1}^{N}\frac{(q^{b+1};q)_{h_j}}{(q;q)_{h_j}}q^{(a+1)h_j}, \tag*{(38)} P N ( h ) = Z N 1 1 ≤ j < k ≤ N ∏ ( q h j − q h k ) 2 j = 1 ∏ N ( q ; q ) h j ( q b + 1 ; q ) h j q ( a + 1 ) h j , (38) where Z ~ N \widetilde{Z}_{N} Z N
Since
lim q → 1 w ( a , b ) ( x ; q ) = x a ( 1 − x ) b , (39) \lim_{q\to1}w^{(a,b)}(x;q)=x^a(1-x)^b, \tag*{(39)} q → 1 lim w ( a , b ) ( x ; q ) = x a ( 1 − x ) b , (39) the classical JUE is recovered from (37) by letting q → 1 q \to1 q → 1 a a a b b b x j ∈ ( 0 , 1 ) x_j \in(0,1) x j ∈ ( 0 , 1 ) (38) by instead keeping the lattice variables h j h_j h j
b → β − 1 , q a + 1 → α 2 , q → 1. (40) b \to\beta-1,\qquad q^{a+1}\to\alpha^2,\qquad q\to1. \tag*{(40)} b → β − 1 , q a + 1 → α 2 , q → 1. (40) For instance, one may take
a = log α 2 log q − 1 , b = β − 1. (41) a=\frac{\log\alpha^2}{\log q}-1,\qquad b=\beta-1. \tag*{(41)} a = log q log α 2 − 1 , b = β − 1. (41) Under this scaling, as q → 1 q \to1 q → 1
( q b + 1 ; q ) h ( q ; q ) h → ( β ) h h ! , q ( a + 1 ) h → α 2 h , q h j − q h k 1 − q → − ( h j − h k ) . \frac{(q^{b+1};q)_h}{(q;q)_h}\to\frac{(\beta)_h}{h!},\qquad q^{(a+1)h}\to\alpha^{2h},\qquad\frac{q^{h_j}-q^{h_k}}{1-q}\to-(h_j-h_k). ( q ; q ) h ( q b + 1 ; q ) h → h ! ( β ) h , q ( a + 1 ) h → α 2 h , 1 − q q h j − q h k → − ( h j − h k ) . The probability mass function (38) therefore converges to the Meixner ensemble (33) . Figure 7 summarises these two degenerations of the q q q
Figure 7. Two degenerations of the q q q q → 1 q \to1 q → 1
Unlike the classical JUE, the asymptotic theory of the q q q q q q [97 ] , does not appear to have been identified in the existing literature. This should be compared with the recent identification of the q q q [30 ] and the q q q [31 ] .
In order to describe the limiting macroscopic behaviour, we first define the empirical spectral distribution
μ N ( d x ) : = 1 N ∑ j = 1 N δ x j . (42) \mu_N(dx):=\frac{1}{N}\sum_{j=1}^{N}\delta_{x_j}. \tag*{(42)} μ N ( d x ) := N 1 j = 1 ∑ N δ x j . (42) As in the classical JUE case, when describing the large-N N N a a a b b b
a = c 1 N + d 1 , b = c 2 N + d 2 . (43) a=c_1N+d_1,\qquad b=c_2N+d_2. \tag*{(43)} a = c 1 N + d 1 , b = c 2 N + d 2 . (43) Here, c 1 , c 2 ≥ 0 c_1,c_2\geq0 c 1 , c 2 ≥ 0 d i > − 1 d_i>-1 d i > − 1 c i = 0 c_i=0 c i = 0 i = 1 , 2 i=1,2 i = 1 , 2 c 1 , c 2 c_1,c_2 c 1 , c 2 d 1 , d 2 d_1,d_2 d 1 , d 2
While the derivation of limiting spectral distributions for classical random matrix ensembles is by now well understood, the corresponding theory for q q q q q q [30 ] to derive the q q q [32 ] to the Al-Salam–Carlitz ensemble, where the limiting spectral measure was shown to coincide with the limiting zero distribution of the associated orthogonal polynomials. More recently, [31 ] combined these algebraic-combinatorial techniques with potential theory [61 ] to establish the global asymptotics of the q q q q q q q q q c 1 , c 2 ≥ 0 c_1,c_2 \ge0 c 1 , c 2 ≥ 0
a = s ( ( 1 − s c 1 + 1 ) ( 1 − s c 1 + c 2 + 1 ) − s c 1 ( 1 − s ) ( 1 − s c 2 + 1 ) 1 − s c 1 + c 2 + 2 ) 2 , b = s ( ( 1 − s c 1 + 1 ) ( 1 − s c 1 + c 2 + 1 ) + s c 1 ( 1 − s ) ( 1 − s c 2 + 1 ) 1 − s c 1 + c 2 + 2 ) 2 , (44) \begin{aligned}
a &= s\left(\frac{\sqrt{(1-s^{c_1+1})(1-s^{c_1+c_2+1})}-\sqrt{s^{c_1}(1-s)(1-s^{c_2+1})}}{1-s^{c_1+c_2+2}}\right)^2,\\
b &= s\left(\frac{\sqrt{(1-s^{c_1+1})(1-s^{c_1+c_2+1})}+\sqrt{s^{c_1}(1-s)(1-s^{c_2+1})}}{1-s^{c_1+c_2+2}}\right)^2, \tag*{(44)}
\end{aligned} a b = s ( 1 − s c 1 + c 2 + 2 ( 1 − s c 1 + 1 ) ( 1 − s c 1 + c 2 + 1 ) − s c 1 ( 1 − s ) ( 1 − s c 2 + 1 ) ) 2 , = s ( 1 − s c 1 + c 2 + 2 ( 1 − s c 1 + 1 ) ( 1 − s c 1 + c 2 + 1 ) + s c 1 ( 1 − s ) ( 1 − s c 2 + 1 ) ) 2 , (44) where
s : = e − λ , λ ∈ ( 0 , ∞ ) . (45) s := e^{-\lambda}, \qquad\lambda\in(0,\infty). \tag*{(45)} s := e − λ , λ ∈ ( 0 , ∞ ) . (45) Then we have the following.
Theorem 3.1 (q q q c 1 , c 2 ≥ 0 c_1,c_2 \ge0 c 1 , c 2 ≥ 0 q = e − λ / N q=e^{-\lambda/N} q = e − λ / N λ ∈ ( 0 , ∞ ) \lambda\in(0,\infty) λ ∈ ( 0 , ∞ ) a a a b b b c 2 > 0 c_2>0 c 2 > 0 λ c \lambda_c λ c
s ( 1 + s c 1 − s c 1 + c 2 + 1 ) = 1 , (46) s(1+s^{c_1}-s^{c_1+c_2+1})=1, \tag*{(46)} s ( 1 + s c 1 − s c 1 + c 2 + 1 ) = 1 , (46) where s s s c 2 = 0 c_2=0 c 2 = 0 λ c = 0 \lambda_c=0 λ c = 0 N → ∞ N\to\infty N → ∞
μ N ( d x ) → ρ ( c 1 , c 2 ) ( x ) d x almost surely , (47) \mu_N(dx)\to\rho^{(c_1,c_2)}(x)\,dx \qquad\text{almost surely}, \tag*{(47)} μ N ( d x ) → ρ ( c 1 , c 2 ) ( x ) d x almost surely , (47) where
ρ ( c 1 , c 2 ) ( x ) : = 1 π λ x arccos ( 1 + s c 1 − ( s − 1 + s c 1 + c 2 + 1 ) x 2 s c 1 ( 1 − x ) ( 1 − s c 2 x ) ) 1 [ a , b ] ( x ) + { 0 , if λ ≤ λ c , 1 λ x 1 [ b , 1 ] ( x ) , if λ > λ c . (48) \rho^{(c_1,c_2)}(x):=\frac{1}{\pi\lambda x}\arccos\left(\frac{1+s^{c_1}-(s^{-1}+s^{c_1+c_2+1})x}{2\sqrt{s^{c_1}(1-x)(1-s^{c_2}x)}}\right)\mathbf{1}_{[a,b]}(x)+\begin{cases}0, & \text{if }\lambda\le\lambda_c,\\ \frac{1}{\lambda x}\mathbf{1}_{[b,1]}(x), & \text{if }\lambda>\lambda_c.\end{cases} \tag*{(48)} ρ ( c 1 , c 2 ) ( x ) := π λ x 1 arccos ( 2 s c 1 ( 1 − x ) ( 1 − s c 2 x ) 1 + s c 1 − ( s − 1 + s c 1 + c 2 + 1 ) x ) 1 [ a , b ] ( x ) + { 0 , λ x 1 1 [ b , 1 ] ( x ) , if λ ≤ λ c , if λ > λ c . (48) Here, a a a b b b
See Figure 8 for a numerical illustration of Theorem 3.1. Rather than sampling the q q q q q q [31 , 32 ] for related discussions.
Figure 8. Histograms of the zero sets of the corresponding little q q q λ < λ c \lambda< \lambda_c λ < λ c λ > λ c \lambda> \lambda_c λ > λ c c 1 = 0 c_1 = 0 c 1 = 0 c 1 > 0 c_1 > 0 c 1 > 0
Remark 4 (Band-to-saturated transition). A notable feature of the limiting density is the presence of a nontrivial phase transition, which we refer to as the band-to-saturated transition . Here the saturated region is the part of the support on which the density in (3.17) attains the upper constraint 1 / ( λ x ) 1/(\lambda x) 1/ ( λ x ) λ > λ c \lambda>\lambda_c λ > λ c [8 , 43 , 61 ] . We also refer to [31 ] for further discussion and intuition explaining why analogous phase transitions arise in q q q
An important aspect of our result, in line with the series of works [30 , 31 , 32 ] , is that both the limiting density and the exact critical transition point can be identified explicitly. Another notable feature is that the transition is absent when c 2 = 0 c_2=0 c 2 = 0
s ( 1 + s c 1 − s c 1 + 1 ) = s + ( 1 − s ) s c 1 + 1 = 1 − ( 1 − s ) ( 1 − s c 1 + 1 ) . s(1+s^{c_1}-s^{c_1+1})=s+(1-s)s^{c_1+1}=1-(1-s)(1-s^{c_1+1}). s ( 1 + s c 1 − s c 1 + 1 ) = s + ( 1 − s ) s c 1 + 1 = 1 − ( 1 − s ) ( 1 − s c 1 + 1 ) . Under the standing assumptions on the parameters, this quantity is strictly less than 1 for 0 < s < 1 0<s<1 0 < s < 1 0 < s ≤ 1 0<s\le1 0 < s ≤ 1 s = 1 s=1 s = 1 λ = 0 \lambda=0 λ = 0 c 2 = 0 c_2=0 c 2 = 0 c 2 > 0 c_2>0 c 2 > 0
Remark 5 (Degenerations to the limiting laws of the JUE and the Meixner ensemble). The limiting law (3.17) recovers the macroscopic limits of both the classical JUE and the Meixner ensemble under two different limiting procedures discussed above.
We first recall that the limiting spectral distribution of the classical JUE is the Wachter distribution [97 ]
ρ J ( x ) = ( c 1 + c 2 + 2 ) ( x − a J ) ( b J − x ) 2 π x ( 1 − x ) (49) \rho_J(x) = (c_1+c_2+2)\frac{\sqrt{(x-a_J)(b_J-x)}}{2\pi x(1-x)} \tag*{(49)} ρ J ( x ) = ( c 1 + c 2 + 2 ) 2 π x ( 1 − x ) ( x − a J ) ( b J − x ) (49) supported on ( a J , b J ) (a_J,b_J) ( a J , b J )
1 ( c 1 + c 2 + 2 ) 2 ( ( c 1 + 1 ) ( c 1 + c 2 + 1 ) ± c 2 + 1 ) 2 . (50) \frac{1}{(c_1+c_2+2)^2}\left(\sqrt{(c_1+1)(c_1+c_2+1)} \pm\sqrt{c_2+1}\right)^2. \tag*{(50)} ( c 1 + c 2 + 2 ) 2 1 ( ( c 1 + 1 ) ( c 1 + c 2 + 1 ) ± c 2 + 1 ) 2 . (50) This distribution is recovered from (3.17) by keeping c 1 c_1 c 1 c 2 c_2 c 2 λ → 0 \lambda\to0 λ → 0 a a a b b b
lim λ → 0 a = a J , lim λ → 0 b = b J , (51) \lim_{\lambda\to0} a = a_J,\qquad\lim_{\lambda\to0} b = b_J, \tag*{(51)} λ → 0 lim a = a J , λ → 0 lim b = b J , (51) and the density converges pointwise on the interior of the limiting support:
lim λ → 0 ρ ( c 1 , c 2 ) ( x ) = ρ J ( x ) , x ∈ ( a J , b J ) . (52) \lim_{\lambda\to0}\rho^{(c_1,c_2)}(x) = \rho_J(x),\qquad x \in(a_J,b_J). \tag*{(52)} λ → 0 lim ρ ( c 1 , c 2 ) ( x ) = ρ J ( x ) , x ∈ ( a J , b J ) . (52) The limiting distribution of the Meixner ensemble is obtained through a different scaling. Let α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) Q ≥ 0 Q \ge0 Q ≥ 0
a M = ( 1 − α Q + 1 ) 2 1 − α 2 , b M = ( 1 + α Q + 1 ) 2 1 − α 2 . (53) a_{\mathrm{M}}=\frac{(1-\alpha\sqrt{Q+1})^2}{1-\alpha^2},\qquad b_{\mathrm{M}}=\frac{(1+\alpha\sqrt{Q+1})^2}{1-\alpha^2}. \tag*{(53)} a M = 1 − α 2 ( 1 − α Q + 1 ) 2 , b M = 1 − α 2 ( 1 + α Q + 1 ) 2 . (53) For a M < u < b M a_{\mathrm{M}}<u<b_{\mathrm{M}} a M < u < b M
ϑ 0 ( u ) = 1 π arctan a M ( b M − u ) b M ( u − a M ) , ϑ Q ( u ) = 1 π arctan ( a M + Q ) ( b M − u ) ( b M + Q ) ( u − a M ) . (54) \vartheta_0(u)=\frac{1}{\pi}\arctan\sqrt{\frac{a_{\mathrm{M}}(b_{\mathrm{M}}-u)}{b_{\mathrm{M}}(u-a_{\mathrm{M}})}},\qquad\vartheta_Q(u)=\frac{1}{\pi}\arctan\sqrt{\frac{(a_{\mathrm{M}}+Q)(b_{\mathrm{M}}-u)}{(b_{\mathrm{M}}+Q)(u-a_{\mathrm{M}})}}. \tag*{(54)} ϑ 0 ( u ) = π 1 arctan b M ( u − a M ) a M ( b M − u ) , ϑ Q ( u ) = π 1 arctan ( b M + Q ) ( u − a M ) ( a M + Q ) ( b M − u ) . (54) The associated limiting density is then
ρ M ( u ) = { ( 1 [ 0 , a M ] ( u ) + ( ϑ 0 ( u ) + ϑ Q ( u ) ) ) 1 ( a M , b M ) ( u ) , if α 2 ( Q + 1 ) < 1 , ( − ϑ 0 ( u ) + ϑ Q ( u ) ) 1 ( a M , b M ) ( u ) , if α 2 ( Q + 1 ) > 1. (55) \rho_{\mathrm{M}}(u)=
\begin{cases}
\left(\mathbb{1}_{[0,a_{\mathrm{M}}]}(u)+(\vartheta_0(u)+\vartheta_Q(u))\right)\mathbb{1}_{(a_{\mathrm{M}},b_{\mathrm{M}})}(u), & \text{if }\alpha^2(Q+1)<1,\\
(-\vartheta_0(u)+\vartheta_Q(u))\mathbb{1}_{(a_{\mathrm{M}},b_{\mathrm{M}})}(u), & \text{if }\alpha^2(Q+1)>1.
\end{cases} \tag*{(55)} ρ M ( u ) = { ( 1 [ 0 , a M ] ( u ) + ( ϑ 0 ( u ) + ϑ Q ( u )) ) 1 ( a M , b M ) ( u ) , ( − ϑ 0 ( u ) + ϑ Q ( u )) 1 ( a M , b M ) ( u ) , if α 2 ( Q + 1 ) < 1 , if α 2 ( Q + 1 ) > 1. (55) The critical case α 2 ( Q + 1 ) = 1 \alpha^2(Q+1)=1 α 2 ( Q + 1 ) = 1 a M = 0 a_{\mathrm{M}}=0 a M = 0 [64 Eqs. (2.19) and (2.20), Section 6] ; see also [40 Propositions 2.1 and 2.2] .
To recover (55) from the q q q x x x
c 1 = − log α 2 λ , c 2 = Q , (56) c_1=-\frac{\log\alpha^2}{\lambda},\qquad c_2=Q, \tag*{(56)} c 1 = − λ log α 2 , c 2 = Q , (56) and introduce the change of variables
u = 1 λ log 1 x , x = e − λ u . (57) u=\frac{1}{\lambda}\log\frac{1}{x},\qquad x=e^{-\lambda u}. \tag*{(57)} u = λ 1 log x 1 , x = e − λ u . (57) Let a λ a_\lambda a λ b λ b_\lambda b λ (44) under this scaling. Then
lim λ → 0 1 λ log 1 b λ = a M , lim λ → 0 1 λ log 1 a λ = b M . (58) \lim_{\lambda\to0}\frac{1}{\lambda}\log\frac{1}{b_\lambda}=a_{\mathrm{M}},\qquad\lim_{\lambda\to0}\frac{1}{\lambda}\log\frac{1}{a_\lambda}=b_{\mathrm{M}}. \tag*{(58)} λ → 0 lim λ 1 log b λ 1 = a M , λ → 0 lim λ 1 log a λ 1 = b M . (58) The reversal of the endpoints is due to the fact that the map u ↦ e − λ u u\mapsto e^{-\lambda u} u ↦ e − λ u
ρ M , λ ( u ) : = λ e − λ u ρ ( − log α 2 / λ , Q ) ( e − λ u ) . (59) \rho_{\mathrm{M},\lambda}(u):=\lambda e^{-\lambda u}\rho^{(-\log\alpha^2/\lambda,Q)}(e^{-\lambda u}). \tag*{(59)} ρ M , λ ( u ) := λ e − λ u ρ ( − l o g α 2 / λ , Q ) ( e − λ u ) . (59) Then it follows from straightforward computations that
lim λ → 0 ρ M , λ ( u ) = ρ M ( u ) , u ∈ ( a M , b M ) . (60) \lim_{\lambda\to0}\rho_{\mathrm{M},\lambda}(u)=\rho_{\mathrm{M}}(u),\qquad u\in(a_{\mathrm{M}},b_{\mathrm{M}}). \tag*{(60)} λ → 0 lim ρ M , λ ( u ) = ρ M ( u ) , u ∈ ( a M , b M ) . (60) Remark 6 (Degenerations to the q q q q q q q q q [31 ] and semicircle laws [30 ] as natural degenerations. Fix λ > 0 \lambda>0 λ > 0 ρ q M P ( c ) \rho_{\mathrm{qMP}}^{(c)} ρ qMP ( c ) q q q [31 Eq. (1.5) and Theorem 1.2] . If c 1 = c c_1=c c 1 = c c 2 → ∞ c_2\to\infty c 2 → ∞ s c 2 → 0 s^{c_2}\to0 s c 2 → 0 (44) and (48) reduce to their q q q
lim c 2 → ∞ ρ ( c , c 2 ) ( x ) = ρ q M P ( c ) ( x ) (61) \lim_{c_2\to\infty}\rho^{(c,c_2)}(x)=\rho_{\mathrm{qMP}}^{(c)}(x) \tag*{(61)} c 2 → ∞ lim ρ ( c , c 2 ) ( x ) = ρ qMP ( c ) ( x ) (61) at every continuity point of the limiting density. The corresponding critical equation reduces to s ( 1 + s c ) = 1 s(1+s^c)=1 s ( 1 + s c ) = 1 q q q q q q [30 Theorem 2.6] is recovered from the special case c = 0 c=0 c = 0 [31 Remark 3] .
Beyond the global limiting distribution, we study the local scaling limits of the q q q [64 ] , where the fluctuation of the passage time in homogeneous LPP was obtained through the largest particle distribution of the Meixner ensemble.
The Riemann–Hilbert theory of discrete orthogonal polynomials [8 ] treats discrete orthogonal polynomial ensembles on finite sets of nodes whose empirical distributions converge to a positive analytic density on a compact interval, together with analytic varying weights. In this setting, one obtains universal limits such as the discrete sine kernel in bands and the Airy kernel at generic band edges. Nevertheless, the hypotheses of this theory do not apply directly to the present q q q q q q
Motivated by this general picture, we derive the relevant scaling limits for the q q q c 1 > 0 c_1>0 c 1 > 0 c 1 = 0 c_1=0 c 1 = 0
To identify the correct edge scaling, we first need the asymptotic behaviour of the limiting density near the relevant edge. A hard edge occurs at x = 0 x=0 x = 0 c 1 = 0 c_1=0 c 1 = 0 a = d 1 a=d_1 a = d 1 q q q
ρ ( 0 , c 2 ) ( x ) = κ 2 π x ( 1 + o ( 1 ) ) , κ : = 2 λ ( 1 − s ) ( 1 − s c 2 + 1 ) s (62) \rho^{(0,c_2)}(x)=\frac{\kappa}{2\pi\sqrt{x}}(1+o(1)),\qquad\kappa:=\frac{2}{\lambda}\sqrt{\frac{(1-s)(1-s^{c_2+1})}{s}} \tag*{(62)} ρ ( 0 , c 2 ) ( x ) = 2 π x κ ( 1 + o ( 1 )) , κ := λ 2 s ( 1 − s ) ( 1 − s c 2 + 1 ) (62) as x ↓ 0 x\downarrow0 x ↓ 0 c 1 > 0 c_1>0 c 1 > 0 a a a (44) is a soft edge. In this case, the q q q
ρ ( c 1 , c 2 ) ( x ) = θ π ( x − a ) 1 / 2 ( 1 + o ( 1 ) ) , x ↓ a , (63) \rho^{(c_1,c_2)}(x)=\frac{\theta}{\pi}(x-a)^{1/2}(1+o(1)),\qquad x\downarrow a, \tag*{(63)} ρ ( c 1 , c 2 ) ( x ) = π θ ( x − a ) 1/2 ( 1 + o ( 1 )) , x ↓ a , (63) where
θ : = { ( 1 − a ) ( 1 − s c 2 b ) − ( 1 − b ) ( 1 − s c 2 a ) λ a ( 1 − a ) ( 1 − s c 2 a ) ( b − a ) , if λ ≤ λ c , ( 1 − a ) ( 1 − s c 2 b ) + ( 1 − b ) ( 1 − s c 2 a ) λ a ( 1 − a ) ( 1 − s c 2 a ) ( b − a ) , if λ > λ c . (64) \theta:=\begin{cases}
\frac{\sqrt{(1-a)(1-s^{c_2}b)}-\sqrt{(1-b)(1-s^{c_2}a)}}{\lambda a\sqrt{(1-a)(1-s^{c_2}a)(b-a)}}, & \text{if }\lambda\le\lambda_c,\\[1.2em]
\frac{\sqrt{(1-a)(1-s^{c_2}b)}+\sqrt{(1-b)(1-s^{c_2}a)}}{\lambda a\sqrt{(1-a)(1-s^{c_2}a)(b-a)}}, & \text{if }\lambda>\lambda_c.
\end{cases} \tag*{(64)} θ := ⎩ ⎨ ⎧ λa ( 1 − a ) ( 1 − s c 2 a ) ( b − a ) ( 1 − a ) ( 1 − s c 2 b ) − ( 1 − b ) ( 1 − s c 2 a ) , λa ( 1 − a ) ( 1 − s c 2 a ) ( b − a ) ( 1 − a ) ( 1 − s c 2 b ) + ( 1 − b ) ( 1 − s c 2 a ) , if λ ≤ λ c , if λ > λ c . (64) Let x min x_{\min} x m i n q q q
Theorem 3.2 (Smallest particle distribution of the q q q Let the parameters a a a b b b (43) . Fix c 2 ≥ 0 c_2\ge0 c 2 ≥ 0 q = e − λ / N q=e^{-\lambda/N} q = e − λ / N λ ∈ ( 0 , ∞ ) \lambda\in(0,\infty) λ ∈ ( 0 , ∞ )
(i) (Hard edge ) Suppose that c 1 = 0 c_1=0 c 1 = 0 s > 0 s>0 s > 0
lim N → ∞ P ( x min > s κ 2 N 2 ) = det ( I − K d 1 B e s s e l ) L 2 ( 0 , s ) (65) \lim_{N\to\infty}\mathbb{P}\left(x_{\min}>\frac{s}{\kappa^2N^2}\right)=\det\left(I-K_{d_1}^{\mathrm{Bessel}}\right)_{L^2(0,s)} \tag*{(65)} N → ∞ lim P ( x m i n > κ 2 N 2 s ) = det ( I − K d 1 Bessel ) L 2 ( 0 , s ) (65) where K d 1 B e s s e l K_{d_1}^{\mathrm{Bessel}} K d 1 Bessel (19) , and κ \kappa κ (62) .
(ii) (Soft edge ) Suppose that c 1 > 0 c_1>0 c 1 > 0 s ∈ R s\in\mathbb{R} s ∈ R
lim N → ∞ P ( x min ≥ a − s ( θ N ) 2 / 3 ) = F T W ( s ) , (66) \lim_{N\to\infty}\mathbb{P}\left(x_{\min}\ge a-\frac{s}{(\theta N)^{2/3}}\right)=F_{\mathrm{TW}}(s), \tag*{(66)} N → ∞ lim P ( x m i n ≥ a − ( θ N ) 2/3 s ) = F TW ( s ) , (66) where F T W F_{\mathrm{TW}} F TW (6) , a a a (44) , and θ \theta θ (64) .
For c 2 > 0 c_2>0 c 2 > 0 d 2 d_2 d 2 R \mathbb{R} R
Theorem 3.2 shows that, although the global limiting distribution of the q q q q q q c 1 c_1 c 1 c 1 = 0 c_1=0 c 1 = 0 c 1 > 0 c_1>0 c 1 > 0
In fact, edge scaling limits for the q q q q q q [30 , 31 , 32 ] , which were concerned primarily with global limiting distributions, by also establishing the local edge statistics.
Through the duality with inhomogeneous LPP established in the following section, the two edge regimes in Theorem 3.2 correspond to the strong and weak inhomogeneity regimes discussed in the previous section. Thus, the q q q
Duality and its consequences
This section is devoted to the duality relations. In Proposition 4.1, we establish an exact duality between inhomogeneous geometric LPP and the q q q q q q
Duality
The duality relation, which is the main structural ingredient of the present paper, is the following. We use the convention N = Z ≥ 0 \mathbb{N} = \mathbb{Z}_{\geq0} N = Z ≥ 0
Proposition 4.1 (Duality between inhomogeneous geometric LPP and the q q q ). Assume that 0 < q < 1 0 < q < 1 0 < q < 1 n ≥ m n \geq m n ≥ m G n , m G_{n,m} G n , m ℓ ≥ 0 \ell\geq0 ℓ ≥ 0
P ( G n , m ≤ ℓ ) = 1 C q ( n , m ; α ) ∑ h ∈ N m max h i ≤ m + ℓ − 1 ∏ 1 ≤ i < j ≤ m ( q h j − q h i ) 2 ∏ j = 1 m ( q ; q ) h j + n − m ( q ; q ) h j α 2 h j . (67) \mathbb{P}(G_{n,m} \leq\ell) = \frac{1}{C_q(n,m;\alpha)} \sum_{\substack{h \in\mathbb{N}^{m}\\ \max h_i \leq m+\ell-1}} \prod_{1 \leq i < j \leq m} (q^{h_j} - q^{h_i})^2 \prod_{j=1}^{m} \frac{(q;q)_{h_j+n-m}}{(q;q)_{h_j}} \alpha^{2h_j}. \tag*{(67)} P ( G n , m ≤ ℓ ) = C q ( n , m ; α ) 1 h ∈ N m m a x h i ≤ m + ℓ − 1 ∑ 1 ≤ i < j ≤ m ∏ ( q h j − q h i ) 2 j = 1 ∏ m ( q ; q ) h j ( q ; q ) h j + n − m α 2 h j . (67) where C q ( n , m ; α ) C_q(n,m;\alpha) C q ( n , m ; α )
Equivalently, let { x j } j = 1 m \{x_j\}_{j=1}^{m} { x j } j = 1 m q q q (37) , with parameters
a = 2 log α log q − 1 , b = n − m . (68) a = \frac{2\log\alpha}{\log q} - 1, \qquad b = n-m. \tag*{(68)} a = log q 2 log α − 1 , b = n − m . (68) Then we have
P ( G n , m ≤ ℓ ) = P ( min 1 ≤ j ≤ m x j ≥ q m + ℓ − 1 ) . (69) \mathbb{P}(G_{n,m} \leq\ell) = \mathbb{P}\left(\min_{1 \leq j \leq m} x_j \geq q^{m+\ell-1}\right). \tag*{(69)} P ( G n , m ≤ ℓ ) = P ( 1 ≤ j ≤ m min x j ≥ q m + ℓ − 1 ) . (69) Remark 7 (Homogeneous limit and the Meixner ensemble). The duality in Proposition 4.1 can be viewed as a q q q [64 Proposition 1.3] , it was shown that, when q = 1 q = 1 q = 1
P ( G n , m ≤ ℓ ) ∣ q = 1 ∝ ∑ h ∈ N m max h i ≤ m + ℓ − 1 ∏ 1 ≤ i < j ≤ m ( h j − h i ) 2 ∏ j = 1 m ( h j + n − m h j ) α 2 h j . (70) \left.\mathbb{P}(G_{n,m} \leq\ell)\right|_{q=1} \propto\sum_{\substack{h \in\mathbb{N}^{m}\\ \max h_i \leq m+\ell-1}} \prod_{1 \leq i < j \leq m} (h_j-h_i)^2 \prod_{j=1}^{m} \binom{h_j+n-m}{h_j} \alpha^{2h_j}. \tag*{(70)} P ( G n , m ≤ ℓ ) ∣ q = 1 ∝ h ∈ N m m a x h i ≤ m + ℓ − 1 ∑ 1 ≤ i < j ≤ m ∏ ( h j − h i ) 2 j = 1 ∏ m ( h j h j + n − m ) α 2 h j . (70) The right-hand side is the distribution function of the largest particle in the Meixner ensemble. As explained in Section 3, the Meixner ensemble arises as a degeneration of the q q q (70) is recovered as a limiting case of (67) .
Let us mention a related appearance of the q q q [14 ] . For the probability measure on plane partitions considered in [14 Eq. (2.1)] , the associated point process is identified in [14 Theorem 2.1] with a discrete Muttalib–Borodin ensemble. The specialisation in [14 Remark 2.3] yields a little q q q [14 Theorem 5.4] .
The correspondence in Proposition 4.1 can also be obtained from [56 Proposition 2] by sending the parameter of the additional column to zero. We give a direct, self-contained derivation below, including the explicit identification of the q q q
Proof of Proposition 4.1. We first fix the notation for Schur polynomials with a prescribed number of variables. A partition is a weakly decreasing sequence λ = ( λ 1 , λ 2 , … ) \lambda= (\lambda_1,\lambda_2,\ldots) λ = ( λ 1 , λ 2 , … ) ℓ ( λ ) : = # { j : λ j > 0 } \ell(\lambda) := \#\{j : \lambda_j > 0\} ℓ ( λ ) := # { j : λ j > 0 } r r r z = ( z 1 , … , z r ) \boldsymbol{z} = (z_1,\ldots,z_r) z = ( z 1 , … , z r )
s λ ( r ) ( z ) = s λ ( z 1 , … , z r ) s_{\lambda}^{(r)}(\boldsymbol{z}) = s_{\lambda}(z_1,\ldots,z_r) s λ ( r ) ( z ) = s λ ( z 1 , … , z r ) for the Schur polynomial in r r r ℓ ( λ ) ≤ r \ell(\lambda) \leq r ℓ ( λ ) ≤ r
s λ ( r ) ( z ) = 1 Δ ( z ) det { z i λ j + r − j } i , j = 1 r , Δ ( z ) = ∏ 1 ≤ i < j ≤ r ( z i − z j ) . (71) s_{\lambda}^{(r)}(\boldsymbol{z}) = \frac{1}{\Delta(\boldsymbol{z})} \det\left\{z_i^{\lambda_j+r-j}\right\}_{i,j=1}^{r}, \qquad\Delta(\boldsymbol{z}) = \prod_{1 \leq i < j \leq r}(z_i-z_j). \tag*{(71)} s λ ( r ) ( z ) = Δ ( z ) 1 det { z i λ j + r − j } i , j = 1 r , Δ ( z ) = 1 ≤ i < j ≤ r ∏ ( z i − z j ) . (71) We use the convention that
s λ ( r ) ( z ) = 0 if l ( λ ) > r . (72) s_{\lambda}^{(r)}(z)=0 \qquad\text{if } l(\lambda)>r. \tag*{(72)} s λ ( r ) ( z ) = 0 if l ( λ ) > r . (72) When the number of variables is clear from the context, we suppress the superscript and simply write s λ ( z ) s_{\lambda}(z) s λ ( z )
Let x ( n ) = ( x 1 , … , x n ) x^{(n)}=(x_1,\ldots,x_n) x ( n ) = ( x 1 , … , x n ) y ( m ) = ( y 1 , … , y m ) y^{(m)}=(y_1,\ldots,y_m) y ( m ) = ( y 1 , … , y m ) 0 ≤ x i y j < 1 0\le x_i y_j<1 0 ≤ x i y j < 1
P ( G n , m ≤ ℓ ) = ∏ i = 1 n ∏ j = 1 m ( 1 − x i y j ) ∑ λ : λ 1 ≤ ℓ s λ ( n ) ( x ( n ) ) s λ ( m ) ( y ( m ) ) . (73) \mathbb{P}(G_{n,m}\le\ell)=\prod_{i=1}^{n}\prod_{j=1}^{m}(1-x_i y_j)\sum_{\lambda:\lambda_1\le\ell}s_{\lambda}^{(n)}(x^{(n)})s_{\lambda}^{(m)}(y^{(m)}). \tag*{(73)} P ( G n , m ≤ ℓ ) = i = 1 ∏ n j = 1 ∏ m ( 1 − x i y j ) λ : λ 1 ≤ ℓ ∑ s λ ( n ) ( x ( n ) ) s λ ( m ) ( y ( m ) ) . (73) where the sum is over all partitions λ \lambda λ λ 1 ≤ ℓ \lambda_1\le\ell λ 1 ≤ ℓ [6 Section 6] and [56 Section 2] . By the above convention, the summand vanishes unless l ( λ ) ≤ min { n , m } l(\lambda)\le\min\{n,m\} l ( λ ) ≤ min { n , m } n ≥ m n\ge m n ≥ m l ( λ ) ≤ m l(\lambda)\le m l ( λ ) ≤ m x i x_i x i y j y_j y j (73) are general admissible parameters; they need not yet be specialised as in (2.3).
From now on, assume that n ≥ m n\ge m n ≥ m
x ( n ) = ( 1 , q , … , q n − 1 ) . x^{(n)}=(1,q,\ldots,q^{n-1}). x ( n ) = ( 1 , q , … , q n − 1 ) . For any partition λ \lambda λ (73) , we have l ( λ ) ≤ m ≤ n l(\lambda)\le m\le n l ( λ ) ≤ m ≤ n
s λ ( n ) ( 1 , q , … , q n − 1 ) = 1 Δ ( 1 , q , … , q n − 1 ) det { ( q λ j + n − j ) i − 1 } i , j = 1 n = 1 Δ ( 1 , q , … , q n − 1 ) ∏ 1 ≤ i < j ≤ n ( q λ j + n − j − q λ i + n − i ) . (74) \begin{aligned}
s_{\lambda}^{(n)}(1,q,\ldots,q^{n-1})=\frac{1}{\Delta(1,q,\ldots,q^{n-1})}\det\left\{\left(q^{\lambda_j+n-j}\right)^{i-1}\right\}_{i,j=1}^{n} \\
&=\frac{1}{\Delta(1,q,\ldots,q^{n-1})}\prod_{1\le i<j\le n}\left(q^{\lambda_j+n-j}-q^{\lambda_i+n-i}\right). \tag*{(74)}
\end{aligned} s λ ( n ) ( 1 , q , … , q n − 1 ) = Δ ( 1 , q , … , q n − 1 ) 1 det { ( q λ j + n − j ) i − 1 } i , j = 1 n = Δ ( 1 , q , … , q n − 1 ) 1 1 ≤ i < j ≤ n ∏ ( q λ j + n − j − q λ i + n − i ) . (74) Here the last equality follows from the Vandermonde determinant identity.
Since only partitions with l ( λ ) ≤ m l(\lambda)\le m l ( λ ) ≤ m λ j = 0 \lambda_j=0 λ j = 0 j > m j>m j > m (74) can be separated into the part involving the first m m m
∏ 1 ≤ i < j ≤ n ( q λ j + n − j − q λ i + n − i ) = ∏ 1 ≤ i < j ≤ m ( q λ j + n − j − q λ i + n − i ) ∏ j = m + 1 n ∏ i = 1 j − 1 ( q n − j − q λ i + n − i ) . (75) \prod_{1\le i<j\le n}\left(q^{\lambda_j+n-j}-q^{\lambda_i+n-i}\right)
=
\prod_{1\le i<j\le m}\left(q^{\lambda_j+n-j}-q^{\lambda_i+n-i}\right)
\prod_{j=m+1}^{n}\prod_{i=1}^{j-1}\left(q^{n-j}-q^{\lambda_i+n-i}\right). \tag*{(75)} 1 ≤ i < j ≤ n ∏ ( q λ j + n − j − q λ i + n − i ) = 1 ≤ i < j ≤ m ∏ ( q λ j + n − j − q λ i + n − i ) j = m + 1 ∏ n i = 1 ∏ j − 1 ( q n − j − q λ i + n − i ) . (75) where we used λ j = 0 \lambda_j=0 λ j = 0 j > m j>m j > m
We next rewrite the second product in terms of the q q q C C C λ \lambda λ ℓ \ell ℓ q , n , m q,n,m q , n , m α \alpha α C C C i = 1 , … , m i=1,\ldots,m i = 1 , … , m
∏ j = m + 1 n ( q n − j − q λ i + n − i ) = q ∑ j = m + 1 n ( n − j ) ∏ j = m + 1 n ( 1 − q λ i + j − i ) = q ( n − m 2 ) ( q ; q ) λ i + n − i ( q ; q ) λ i + m − i . (76) \prod_{j=m+1}^{n}\left(q^{n-j}-q^{\lambda_i+n-i}\right)
=q^{\sum_{j=m+1}^{n}(n-j)}\prod_{j=m+1}^{n}\left(1-q^{\lambda_i+j-i}\right)
=q^{\binom{n-m}{2}}\frac{(q;q)_{\lambda_i+n-i}}{(q;q)_{\lambda_i+m-i}}. \tag*{(76)} j = m + 1 ∏ n ( q n − j − q λ i + n − i ) = q ∑ j = m + 1 n ( n − j ) j = m + 1 ∏ n ( 1 − q λ i + j − i ) = q ( 2 n − m ) ( q ; q ) λ i + m − i ( q ; q ) λ i + n − i . (76) The remaining factors in the second product of (75) , namely those with i > m i>m i > m λ \lambda λ
∏ j = m + 1 n ∏ i = 1 j − 1 ( q n − j − q λ i + n − i ) = C ∏ i = 1 m ( q ; q ) λ i + n − i ( q ; q ) λ i + m − i . (77) \prod_{j=m+1}^{n}\prod_{i=1}^{j-1}\left(q^{n-j}-q^{\lambda_i+n-i}\right)
=C\prod_{i=1}^{m}\frac{(q;q)_{\lambda_i+n-i}}{(q;q)_{\lambda_i+m-i}}. \tag*{(77)} j = m + 1 ∏ n i = 1 ∏ j − 1 ( q n − j − q λ i + n − i ) = C i = 1 ∏ m ( q ; q ) λ i + m − i ( q ; q ) λ i + n − i . (77) Moreover,
∏ 1 ≤ i < j ≤ m ( q λ j + n − j − q λ i + n − i ) = C ∏ 1 ≤ i < j ≤ m ( q λ j + m − j − q λ i + m − i ) . (78) \prod_{1\le i<j\le m}\left(q^{\lambda_j+n-j}-q^{\lambda_i+n-i}\right)
=C\prod_{1\le i<j\le m}\left(q^{\lambda_j+m-j}-q^{\lambda_i+m-i}\right). \tag*{(78)} 1 ≤ i < j ≤ m ∏ ( q λ j + n − j − q λ i + n − i ) = C 1 ≤ i < j ≤ m ∏ ( q λ j + m − j − q λ i + m − i ) . (78) Combining (74) , (75) , (77) , and (78) with the analogous formula for s λ ( m ) ( 1 , q , … , q m − 1 ) s_{\lambda}^{(m)}(1,q,\ldots,q^{m-1}) s λ ( m ) ( 1 , q , … , q m − 1 )
s λ ( n ) ( 1 , q , … , q n − 1 ) s λ ( m ) ( 1 , q , … , q m − 1 ) = C ∏ 1 ≤ i < j ≤ m ( q λ j + m − j − q λ i + m − i ) 2 ∏ j = 1 m ( q ; q ) λ j + n − j ( q ; q ) λ j + m − j . (79) s_{\lambda}^{(n)}(1,q,\ldots,q^{n-1})s_{\lambda}^{(m)}(1,q,\ldots,q^{m-1})
=C\prod_{1\le i<j\le m}\left(q^{\lambda_j+m-j}-q^{\lambda_i+m-i}\right)^2\prod_{j=1}^{m}\frac{(q;q)_{\lambda_j+n-j}}{(q;q)_{\lambda_j+m-j}}. \tag*{(79)} s λ ( n ) ( 1 , q , … , q n − 1 ) s λ ( m ) ( 1 , q , … , q m − 1 ) = C 1 ≤ i < j ≤ m ∏ ( q λ j + m − j − q λ i + m − i ) 2 j = 1 ∏ m ( q ; q ) λ j + m − j ( q ; q ) λ j + n − j . (79) Now we put the parameters x i = α q i − 1 , y j = α q j − 1 x_i=\alpha q^{i-1},y_j=\alpha q^{j-1} x i = α q i − 1 , y j = α q j − 1
s λ ( α x ) = α ∣ λ ∣ s λ ( x ) (80) s_{\lambda}(\alpha x)=\alpha^{|\lambda|}s_{\lambda}(x) \tag*{(80)} s λ ( α x ) = α ∣ λ ∣ s λ ( x ) (80) and absorbing the prefactor ∏ i = 1 n ∏ j = 1 m ( 1 − α 2 q i + j − 2 ) \prod_{i=1}^{n}\prod_{j=1}^{m}(1-\alpha^{2}q^{i+j-2}) ∏ i = 1 n ∏ j = 1 m ( 1 − α 2 q i + j − 2 ) (73) into the constant C C C
P ( G n , m ≤ ℓ ) = C ∑ λ : λ 1 ≤ ℓ l ( λ ) ≤ m α 2 ∣ λ ∣ ∏ 1 ≤ i < j ≤ m ( q λ j + m − j − q λ i + m − i ) 2 ∏ j = 1 m ( q ; q ) λ j + n − j ( q ; q ) λ j + m − j . (81) \mathbb{P}(G_{n,m}\leq\ell)=C\sum_{\substack{\lambda:\lambda_{1}\leq\ell\\l(\lambda)\leq m}}\alpha^{2|\lambda|}\prod_{1\leq i<j\leq m}(q^{\lambda_{j}+m-j}-q^{\lambda_{i}+m-i})^{2}\prod_{j=1}^{m}\frac{(q;q)_{\lambda_{j}+n-j}}{(q;q)_{\lambda_{j}+m-j}}. \tag*{(81)} P ( G n , m ≤ ℓ ) = C λ : λ 1 ≤ ℓ l ( λ ) ≤ m ∑ α 2∣ λ ∣ 1 ≤ i < j ≤ m ∏ ( q λ j + m − j − q λ i + m − i ) 2 j = 1 ∏ m ( q ; q ) λ j + m − j ( q ; q ) λ j + n − j . (81) We now make the change of variables
h j = λ j + m − j , j = 1 , … , m . h_{j}=\lambda_{j}+m-j,\qquad j=1,\ldots,m. h j = λ j + m − j , j = 1 , … , m . Then (81) becomes
P ( G n , m ≤ ℓ ) = C ∑ h ∈ N m max h i ≤ m + ℓ − 1 ∏ 1 ≤ i < j ≤ m ( q h j − q h i ) 2 ∏ j = 1 m ( q ; q ) h j + n − m ( q ; q ) h j α 2 h j . (82) \mathbb{P}(G_{n,m}\leq\ell)=C\sum_{\substack{h\in\mathbb{N}^{m}\\\max h_{i}\leq m+\ell-1}}\prod_{1\leq i<j\leq m}(q^{h_{j}}-q^{h_{i}})^{2}\prod_{j=1}^{m}\frac{(q;q)_{h_{j}+n-m}}{(q;q)_{h_{j}}}\alpha^{2h_{j}}. \tag*{(82)} P ( G n , m ≤ ℓ ) = C h ∈ N m m a x h i ≤ m + ℓ − 1 ∑ 1 ≤ i < j ≤ m ∏ ( q h j − q h i ) 2 j = 1 ∏ m ( q ; q ) h j ( q ; q ) h j + n − m α 2 h j . (82) Letting ℓ → ∞ \ell\to\infty ℓ → ∞ P ( G n , m < ∞ ) = 1 \mathbb{P}(G_{n,m}<\infty)=1 P ( G n , m < ∞ ) = 1 (67) .
It remains to identify the right-hand side of (67) with the q q q q q q (36) on the q q q x j = q h j x_{j}=q^{h_{j}} x j = q h j Z ≥ 0 m \mathbb{Z}_{\geq0}^{m} Z ≥ 0 m
∏ 1 ≤ i < j ≤ m ( q h i − q h j ) 2 ∏ j = 1 m q h j w ( a , b ) ( q h j ; q ) . \prod_{1\leq i<j\leq m}(q^{h_{i}}-q^{h_{j}})^{2}\prod_{j=1}^{m}q^{h_{j}}w^{(a,b)}(q^{h_{j}};q). 1 ≤ i < j ≤ m ∏ ( q h i − q h j ) 2 j = 1 ∏ m q h j w ( a , b ) ( q h j ; q ) . For x = q h x=q^{h} x = q h h ∈ N h\in\mathbb{N} h ∈ N q q q (35) can be written in finite q q q
( a ; q ) ∞ ( a q r ; q ) ∞ = ( a ; q ) r , r ∈ Z ≥ 0 , \frac{(a;q)_{\infty}}{(aq^{r};q)_{\infty}}=(a;q)_{r},\qquad r\in\mathbb{Z}_{\geq0}, ( a q r ; q ) ∞ ( a ; q ) ∞ = ( a ; q ) r , r ∈ Z ≥ 0 , the corresponding one-particle mass at x = q h x=q^{h} x = q h
q h w ( a , b ) ( q h ; q ) = q ( a + 1 ) h ( q h + 1 ; q ) b . (83) q^{h}w^{(a,b)}(q^{h};q)=q^{(a+1)h}(q^{h+1};q)_{b}. \tag*{(83)} q h w ( a , b ) ( q h ; q ) = q ( a + 1 ) h ( q h + 1 ; q ) b . (83) With the parameter matching (68) , we have
q ( a + 1 ) h = α 2 h , ( q h + 1 ; q ) n − m = ( q ; q ) h + n − m ( q ; q ) h . q^{(a+1)h}=\alpha^{2h},\qquad(q^{h+1};q)_{n-m}=\frac{(q;q)_{h+n-m}}{(q;q)_{h}}. q ( a + 1 ) h = α 2 h , ( q h + 1 ; q ) n − m = ( q ; q ) h ( q ; q ) h + n − m . Thus the one-particle weight in (67) is precisely the q q q (68) . Since 0 < q < 1 0<q<1 0 < q < 1 x j = q h j x_{j}=q^{h_{j}} x j = q h j (67) is therefore equivalent to (69) . This completes the proof. □
We next record the continuum counterpart of Proposition 4.1 . In the homogeneous case, Johansson identified exponential LPP with the Wishart (Laguerre) ensemble whose asymptotic analysis is substantially simpler than that of the discrete Meixner ensemble associated with geometric LPP. A similar phenomenon occurs in the present inhomogeneous setting. While the geometric model is naturally related to the q q q
We denote the smallest eigenvalue of the JUE defined in (32) by y min ( N ) ( a , b ) : = min 1 ≤ j ≤ N y j y_{\min}^{(N)}(a,b):=\min_{1\leq j\leq N}y_{j} y m i n ( N ) ( a , b ) := min 1 ≤ j ≤ N y j Proposition 4.1 .
Corollary 4.2 (Duality between inhomogeneous exponential LPP and the JUE). Assume that n ≥ m n\geq m n ≥ m η , λ > 0 \eta,\lambda>0 η , λ > 0 H n , m H_{n,m} H n , m (28) , where the parameters t k t_{k} t k (27) . Let β \beta β t k t_{k} t k ℓ ≥ 0 \ell\geq0 ℓ ≥ 0
P ( H n , m ≤ ℓ ) = P ( y min ( m ) ( 2 β + 1 , n − m ) ≥ e − λ ℓ ) . (84) \mathbb{P}(H_{n,m}\leq\ell)=\mathbb{P}\left(y_{\min}^{(m)}(2\beta+1,n-m)\geq e^{-\lambda\ell}\right). \tag*{(84)} P ( H n , m ≤ ℓ ) = P ( y m i n ( m ) ( 2 β + 1 , n − m ) ≥ e − λ ℓ ) . (84) Proof. Let q = e − ϵ λ q=e^{-\epsilon\lambda} q = e − ϵ λ α = e − ϵ η \alpha=e^{-\epsilon\eta} α = e − ϵη ϵ > 0 \epsilon>0 ϵ > 0 x i x_{i} x i y j y_{j} y j (3) . Then as ϵ → 0 \epsilon\to0 ϵ → 0
x i y j = α 2 q i + j − 2 = exp ( − ϵ ( 2 η + λ ( i + j − 2 ) ) ) = 1 − ϵ ( 2 η + λ ( i + j − 2 ) ) + O ( ϵ 2 ) . x_{i}y_{j}=\alpha^{2}q^{i+j-2}=\exp\left(-\epsilon(2\eta+\lambda(i+j-2))\right)=1-\epsilon(2\eta+\lambda(i+j-2))+O(\epsilon^{2}). x i y j = α 2 q i + j − 2 = exp ( − ϵ ( 2 η + λ ( i + j − 2 )) ) = 1 − ϵ ( 2 η + λ ( i + j − 2 )) + O ( ϵ 2 ) . :::`
Recalling that ω i j \omega_{ij} ω ij x i y j x_i y_j x i y j ϵ → 0 \epsilon\to0 ϵ → 0 ϵ ω i j \epsilon\omega_{ij} ϵ ω ij t i + t j t_i+t_j t i + t j
Note that
( q ; q ) h j + n − m ( q ; q ) h j = ∏ k = 1 n − m ( 1 − q h j + k ) . \frac{(q;q)_{h_j+n-m}}{(q;q)_{h_j}}=\prod_{k=1}^{n-m}(1-q^{h_j+k}). ( q ; q ) h j ( q ; q ) h j + n − m = k = 1 ∏ n − m ( 1 − q h j + k ) . Since G n , m G_{n,m} G n , m w i , j w_{i,j} w i , j
P ( H n , m ≤ ℓ ) = lim ϵ → 0 P ( G n , m ≤ ℓ / ϵ ) ∣ q = e − ϵ λ , α = e − ϵ η = lim ϵ → 0 [ 1 C q ( n , m ; α ) ∑ h ∈ N m max { h i } ≤ m + ℓ / ϵ − 1 ∏ 1 ≤ i < j ≤ m ( q h j − q h i ) 2 ∏ j = 1 m ( q ; q ) h j + n − m ( q ; q ) h j α 2 h j ] q = e − ϵ λ , α = e − ϵ η = lim ϵ → 0 [ 1 C q ( n , m ; α ) ∑ h ∈ N m max { h i } ≤ m + ℓ / ϵ − 1 ∏ 1 ≤ i < j ≤ m ( e − ϵ λ h j − e − ϵ λ h i ) 2 ∏ j = 1 m ( ∏ k = 1 n − m ( 1 − e − ϵ λ ( h j + k ) ) ) e − 2 ϵ η h j ] = lim ϵ → 0 [ 1 C q ( n , m ; α ) ∑ x ∈ ϵ N m max { x i } ≤ ℓ + ϵ ( m − 1 ) ∏ 1 ≤ i < j ≤ m ( e − λ x j − e − λ x i ) 2 ∏ j = 1 m ( ∏ k = 1 n − m ( 1 − e − λ x j − ϵ λ k ) ) e − 2 η x j ] , \begin{aligned}
\mathbb{P}(H_{n,m}\le\ell)
&=\lim_{\epsilon\to0}\left.\mathbb{P}(G_{n,m}\le\ell/\epsilon)\right|_{q=e^{-\epsilon\lambda},\alpha=e^{-\epsilon\eta}}\\
&=\lim_{\epsilon\to0}\left[\frac{1}{C_q(n,m;\alpha)}
\sum_{\substack{h\in\mathbb{N}^m\\ \max\{h_i\}\le m+\ell/\epsilon-1}}
\prod_{1\le i<j\le m}(q^{h_j}-q^{h_i})^2
\prod_{j=1}^m\frac{(q;q)_{h_j+n-m}}{(q;q)_{h_j}}\alpha^{2h_j}\right]_{q=e^{-\epsilon\lambda},\alpha=e^{-\epsilon\eta}}\\
&=\lim_{\epsilon\to0}\left[\frac{1}{C_q(n,m;\alpha)}
\sum_{\substack{h\in\mathbb{N}^m\\ \max\{h_i\}\le m+\ell/\epsilon-1}}
\prod_{1\le i<j\le m}(e^{-\epsilon\lambda h_j}-e^{-\epsilon\lambda h_i})^2
\prod_{j=1}^m\left(\prod_{k=1}^{n-m}(1-e^{-\epsilon\lambda(h_j+k)})\right)e^{-2\epsilon\eta h_j}\right]\\
&=\lim_{\epsilon\to0}\left[\frac{1}{C_q(n,m;\alpha)}
\sum_{\substack{x\in\epsilon\mathbb{N}^m\\ \max\{x_i\}\le\ell+\epsilon(m-1)}}
\prod_{1\le i<j\le m}(e^{-\lambda x_j}-e^{-\lambda x_i})^2
\prod_{j=1}^m\left(\prod_{k=1}^{n-m}(1-e^{-\lambda x_j-\epsilon\lambda k})\right)e^{-2\eta x_j}\right],
\end{aligned} P ( H n , m ≤ ℓ ) = ϵ → 0 lim P ( G n , m ≤ ℓ / ϵ ) ∣ q = e − ϵ λ , α = e − ϵη = ϵ → 0 lim C q ( n , m ; α ) 1 h ∈ N m m a x { h i } ≤ m + ℓ / ϵ − 1 ∑ 1 ≤ i < j ≤ m ∏ ( q h j − q h i ) 2 j = 1 ∏ m ( q ; q ) h j ( q ; q ) h j + n − m α 2 h j q = e − ϵ λ , α = e − ϵη = ϵ → 0 lim C q ( n , m ; α ) 1 h ∈ N m m a x { h i } ≤ m + ℓ / ϵ − 1 ∑ 1 ≤ i < j ≤ m ∏ ( e − ϵ λ h j − e − ϵ λ h i ) 2 j = 1 ∏ m ( k = 1 ∏ n − m ( 1 − e − ϵ λ ( h j + k ) ) ) e − 2 ϵη h j = ϵ → 0 lim C q ( n , m ; α ) 1 x ∈ ϵ N m m a x { x i } ≤ ℓ + ϵ ( m − 1 ) ∑ 1 ≤ i < j ≤ m ∏ ( e − λ x j − e − λ x i ) 2 j = 1 ∏ m ( k = 1 ∏ n − m ( 1 − e − λ x j − ϵ λk ) ) e − 2 η x j , where x j = ϵ h j x_j=\epsilon h_j x j = ϵ h j
P ( H n , m ≤ ℓ ) ∝ ∫ [ 0 , ℓ ] m ∏ 1 ≤ i < j ≤ m ( e − λ x j − e − λ x i ) 2 ∏ j = 1 m ( 1 − e − λ x j ) n − m e − 2 η x j d x j . (85) \mathbb{P}(H_{n,m}\le\ell)\propto
\int_{[0,\ell]^m}\prod_{1\le i<j\le m}(e^{-\lambda x_j}-e^{-\lambda x_i})^2
\prod_{j=1}^m(1-e^{-\lambda x_j})^{n-m}e^{-2\eta x_j}\,dx_j. \tag*{(85)} P ( H n , m ≤ ℓ ) ∝ ∫ [ 0 , ℓ ] m 1 ≤ i < j ≤ m ∏ ( e − λ x j − e − λ x i ) 2 j = 1 ∏ m ( 1 − e − λ x j ) n − m e − 2 η x j d x j . (85) where the multiplicative normalisation constant is independent of ℓ \ell ℓ y j = e − λ x j y_j=e^{-\lambda x_j} y j = e − λ x j
x j = − 1 λ log y j , d x j = − 1 λ y j d y j , (86) x_j=-\frac{1}{\lambda}\log y_j,\qquad dx_j=-\frac{1}{\lambda y_j}\,dy_j, \tag*{(86)} x j = − λ 1 log y j , d x j = − λ y j 1 d y j , (86) we have
P ( H n , m ≤ ℓ ) ∝ ∫ [ e − λ ℓ , 1 ] m ∏ 1 ≤ i < j ≤ m ( y j − y i ) 2 ∏ j = 1 m y j 2 β + 1 ( 1 − y j ) n − m d y j . (87) \mathbb{P}(H_{n,m}\le\ell)\propto
\int_{[e^{-\lambda\ell},1]^m}\prod_{1\le i<j\le m}(y_j-y_i)^2
\prod_{j=1}^m y_j^{2\beta+1}(1-y_j)^{n-m}\,dy_j. \tag*{(87)} P ( H n , m ≤ ℓ ) ∝ ∫ [ e − λ ℓ , 1 ] m 1 ≤ i < j ≤ m ∏ ( y j − y i ) 2 j = 1 ∏ m y j 2 β + 1 ( 1 − y j ) n − m d y j . (87) Thus we conclude (84) , where the normalisation constant is fixed by letting ℓ → ∞ \ell\to\infty ℓ → ∞
Remark 8 (Extended Wishart correspondence). A related matrix model appears in [20 ] , where inhomogeneous exponential LPP is connected to the so-called extended Wishart ensemble. Let X = ( X i j ) X=(X_{ij}) X = ( X ij ) N × p N\times p N × p
X i j ∼ N C ( 0 , 1 t i + t j ) . (88) X_{ij}\sim\mathcal{N}_{\mathbb{C}}\left(0,\frac{1}{t_i+t_j}\right). \tag*{(88)} X ij ∼ N C ( 0 , t i + t j 1 ) . (88) Thus X X X t i + t j t_i+t_j t i + t j [28 ] . The matrix X X ∗ XX^* X X ∗
It was shown in [20 Theorem 1] that the last passage time is related to the largest eigenvalue of this matrix. More precisely, if y ~ max \widetilde{y}_{\max} y m a x X X ∗ XX^* X X ∗
P ( H p , N ≤ x ) = P ( y ~ max ≤ x ) . (89) \mathbb{P}(H_{p,N}\le x)=\mathbb{P}(\widetilde{y}_{\max}\le x). \tag*{(89)} P ( H p , N ≤ x ) = P ( y m a x ≤ x ) . (89) The change of variables used in (86) transforms this extended Wishart description into the Jacobi form appearing in Corollary 4.2. Thus Corollary 4.2 can also be recovered from [20 ] .
Proof of Theorems 2.1, 2.2 and 2.3
In this subsection, we derive the LPP results from the duality relations established in the previous subsection, namely Proposition 4.1 and Corollary 4.2. For this purpose, we take the q q q Theorems 3.1 and 3.2 ) as given and use the duality relations to deduce the corresponding LPP results (Theorems 2.1 , 2.2 , and 2.3 ). Thus, the proofs in this subsection are essentially consequences of the duality, while the underlying asymptotic analysis of the q q q
Before turning to the proofs, we note two minor technicalities. Since the geometric LPP random variables are integer-valued, the exact threshold arising from the duality is q N + ℓ − 1 q^{N+\ell-1} q N + ℓ − 1 Theorem 3.2 . This discrepancy is handled by a squeezing argument using monotonicity and continuity of the limiting Fredholm determinants. Moreover, for γ > 1 \gamma>1 γ > 1
b N = ( γ − 1 ) N + ( ⌊ γ N ⌋ − γ N ) , ⌊ γ N ⌋ − γ N ∈ [ − 1 , 0 ] . b_N=(\gamma-1)N+\left(\lfloor\gamma N\rfloor-\gamma N\right),\qquad\lfloor\gamma N\rfloor-\gamma N\in[-1,0]. b N = ( γ − 1 ) N + ( ⌊ γ N ⌋ − γ N ) , ⌊ γ N ⌋ − γ N ∈ [ − 1 , 0 ] . The uniformity in d 2 d_2 d 2 Theorem 3.2 therefore allows us to apply that theorem with c 2 = γ − 1 c_2=\gamma-1 c 2 = γ − 1
We first apply Proposition 4.1 to prove Theorems 2.1 and 2.2 , assuming the q q q Theorems 3.1 and 3.2 .
Proof of Theorems 2.1 and 2.2. By Proposition 4.1 , for each integer ℓ ≥ 0 \ell\ge0 ℓ ≥ 0
P ( G ⌊ γ N ⌋ , N ≤ ℓ ) = P ( x min ( N ) ≥ q N + ℓ − 1 ) . (90) \mathbb{P}\left(G_{\lfloor\gamma N\rfloor,N}\le\ell\right)=\mathbb{P}\left(x_{\min}^{(N)}\ge q^{N+\ell-1}\right). \tag*{(90)} P ( G ⌊ γ N ⌋ , N ≤ ℓ ) = P ( x m i n ( N ) ≥ q N + ℓ − 1 ) . (90) where x min ( N ) x_{\min}^{(N)} x m i n ( N ) q q q N N N
a N = 2 log α log q − 1 = − 2 log α λ N − 1 , b N = ⌊ γ N ⌋ − N . a_N=\frac{2\log\alpha}{\log q}-1=-\frac{2\log\alpha}{\lambda}N-1,\qquad b_N=\lfloor\gamma N\rfloor-N. a N = log q 2 log α − 1 = − λ 2 log α N − 1 , b N = ⌊ γ N ⌋ − N . We first prove Theorem 2.1 . For notational simplicity, throughout the proof we write
ω = ω ( γ , α , λ ) , σ = σ ( γ , α , λ ) , θ = θ γ , α , λ . \omega=\omega(\gamma,\alpha,\lambda),\qquad\sigma=\sigma(\gamma,\alpha,\lambda),\qquad\theta=\theta_{\gamma,\alpha,\lambda}. ω = ω ( γ , α , λ ) , σ = σ ( γ , α , λ ) , θ = θ γ , α , λ . Under the assumptions of Theorem 2.1 , the corresponding q q q Theorem 3.2 , with macroscopic parameters
c 1 = − 2 log α λ > 0 , c 2 = γ − 1. (91) c_1=-\frac{2\log\alpha}{\lambda}>0,\qquad c_2=\gamma-1. \tag*{(91)} c 1 = − λ 2 log α > 0 , c 2 = γ − 1. (91) Then the left endpoint in (44) is precisely a a a (9) , the right endpoint is b b b (10) , and the soft edge constant in (64) is θ \theta θ (11) . Moreover, s ( 1 + s c 1 − s c 1 + c 2 + 1 ) = Ξ γ , α , λ s(1+s^{c_1}-s^{c_1+c_2+1})=\Xi_{\gamma,\alpha,\lambda} s ( 1 + s c 1 − s c 1 + c 2 + 1 ) = Ξ γ , α , λ Theorem 3.2 (ii) gives, for each r ∈ R r\in\mathbb{R} r ∈ R
lim N → ∞ P ( x min ( N ) ≥ a − r ( θ N ) 2 / 3 ) = F T W ( r ) . (92) \lim_{N\to\infty}\mathbb{P}\left(x_{\min}^{(N)}\ge a-\frac{r}{(\theta N)^{2/3}}\right)=F_{\mathrm{TW}}(r). \tag*{(92)} N → ∞ lim P ( x m i n ( N ) ≥ a − ( θ N ) 2/3 r ) = F TW ( r ) . (92) Moreover, by the definition of a a a ω \omega ω
a = e − λ ( 1 + ω ) . (93) a=e^{-\lambda(1+\omega)}. \tag*{(93)} a = e − λ ( 1 + ω ) . (93) We first prove the law of large numbers. Fix ε > 0 \varepsilon>0 ε > 0 ℓ N + = ⌊ N ( ω + ε ) ⌋ \ell_N^+=\lfloor N(\omega+\varepsilon)\rfloor ℓ N + = ⌊ N ( ω + ε )⌋ (93) ,
q N + ℓ N + − 1 = exp ( − λ N ( N + ℓ N + − 1 ) ) = a e − λ ε + o ( 1 ) . q^{N+\ell_N^+-1}=\exp\left(-\frac{\lambda}{N}(N+\ell_N^+-1)\right)=ae^{-\lambda\varepsilon+o(1)}. q N + ℓ N + − 1 = exp ( − N λ ( N + ℓ N + − 1 ) ) = a e − λ ε + o ( 1 ) . Since e − λ ε < 1 e^{-\lambda\varepsilon}<1 e − λ ε < 1 a a a N N N R > 0 R>0 R > 0 N N N
q N + ℓ N + − 1 ≤ a − R ( θ N ) 2 / 3 . q^{N+\ell_N^+-1}\le a-\frac{R}{(\theta N)^{2/3}}. q N + ℓ N + − 1 ≤ a − ( θ N ) 2/3 R . Then by (90) and (92) , we obtain
lim inf N → ∞ P ( G ⌊ γ N ⌋ , N ≤ ℓ N + ) ≥ lim N → ∞ P ( x min ( N ) ≥ a − R ( θ N ) 2 / 3 ) = F T W ( R ) . \liminf_{N\to\infty}\mathbb{P}\left(G_{\lfloor\gamma N\rfloor,N}\le\ell_N^+\right)\ge\lim_{N\to\infty}\mathbb{P}\left(x_{\min}^{(N)}\ge a-\frac{R}{(\theta N)^{2/3}}\right)=F_{\mathrm{TW}}(R). N → ∞ lim inf P ( G ⌊ γ N ⌋ , N ≤ ℓ N + ) ≥ N → ∞ lim P ( x m i n ( N ) ≥ a − ( θ N ) 2/3 R ) = F TW ( R ) . Letting R → ∞ R\to\infty R → ∞
lim N → ∞ P ( G ⌊ γ N ⌋ , N ≤ N ( ω + ε ) ) = 1. \lim_{N\to\infty}\mathbb{P}\left(G_{\lfloor\gamma N\rfloor,N}\le N(\omega+\varepsilon)\right)=1. N → ∞ lim P ( G ⌊ γ N ⌋ , N ≤ N ( ω + ε ) ) = 1. Similarly, it follows that
lim N → ∞ P ( G ⌊ γ N ⌋ , N ≤ N ( ω − ε ) ) = 0. \lim_{N \to\infty} \mathbb{P}\left(G_{\lfloor\gamma N\rfloor,N} \leq N(\omega-\varepsilon)\right)=0. N → ∞ lim P ( G ⌊ γ N ⌋ , N ≤ N ( ω − ε ) ) = 0. Combining the two estimates gives (2.14).
It remains to prove the fluctuation limit. Fix s ∈ R s \in\mathbb{R} s ∈ R
ℓ N ( s ) : = ⌊ N ω + σ N 1 / 3 s ⌋ = N ω + σ N 1 / 3 s + O ( 1 ) . \ell_N(s):=\lfloor N\omega+\sigma N^{1/3}s\rfloor=N\omega+\sigma N^{1/3}s+O(1). ℓ N ( s ) := ⌊ N ω + σ N 1/3 s ⌋ = N ω + σ N 1/3 s + O ( 1 ) . It follows from (4.27) and the definition (2.13) that
q N + ℓ N ( s ) − 1 = a − λ a σ s N − 2 / 3 + o ( N − 2 / 3 ) = a − s ( θ N ) 2 / 3 + o ( N − 2 / 3 ) . (94) q^{N+\ell_N(s)-1}=\mathfrak{a}-\lambda\mathfrak{a}\sigma sN^{-2/3}+o(N^{-2/3})=\mathfrak{a}-\frac{s}{(\theta N)^{2/3}}+o(N^{-2/3}). \tag*{(94)} q N + ℓ N ( s ) − 1 = a − λ a σ s N − 2/3 + o ( N − 2/3 ) = a − ( θ N ) 2/3 s + o ( N − 2/3 ) . (94) For any δ > 0 \delta>0 δ > 0 N N N
a − s + δ ( θ N ) 2 / 3 ≤ q N + ℓ N ( s ) − 1 ≤ a − s − δ ( θ N ) 2 / 3 . \mathfrak{a}-\frac{s+\delta}{(\theta N)^{2/3}}\leq q^{N+\ell_N(s)-1}\leq\mathfrak{a}-\frac{s-\delta}{(\theta N)^{2/3}}. a − ( θ N ) 2/3 s + δ ≤ q N + ℓ N ( s ) − 1 ≤ a − ( θ N ) 2/3 s − δ . Using monotonicity and (4.26), we obtain
F T W ( s − δ ) ≤ lim inf N → ∞ P ( x min ( N ) ≥ q N + ℓ N ( s ) − 1 ) , lim sup N → ∞ P ( x min ( N ) ≥ q N + ℓ N ( s ) − 1 ) ≤ F T W ( s + δ ) . F_{\mathrm{TW}}(s-\delta)\leq\liminf_{N\to\infty}\mathbb{P}\left(x_{\min}^{(N)}\geq q^{N+\ell_N(s)-1}\right),\qquad
\limsup_{N\to\infty}\mathbb{P}\left(x_{\min}^{(N)}\geq q^{N+\ell_N(s)-1}\right)\leq F_{\mathrm{TW}}(s+\delta). F TW ( s − δ ) ≤ N → ∞ lim inf P ( x m i n ( N ) ≥ q N + ℓ N ( s ) − 1 ) , N → ∞ lim sup P ( x m i n ( N ) ≥ q N + ℓ N ( s ) − 1 ) ≤ F TW ( s + δ ) . Since F T W F_{\mathrm{TW}} F TW δ ↓ 0 \delta\downarrow0 δ ↓ 0
lim N → ∞ P ( x min ( N ) ≥ q N + ℓ N ( s ) − 1 ) = F T W ( s ) . \lim_{N\to\infty}\mathbb{P}\left(x_{\min}^{(N)}\geq q^{N+\ell_N(s)-1}\right)=F_{\mathrm{TW}}(s). N → ∞ lim P ( x m i n ( N ) ≥ q N + ℓ N ( s ) − 1 ) = F TW ( s ) . By the duality (4.24), we obtain (2.15), which completes the proof of Theorem 2.1.
Next, we prove Theorem 2.2. Since log q = − λ / N \log q=-\lambda/N log q = − λ / N log α N = − η / N \log\alpha_N=-\eta/N log α N = − η / N q q q
c 1 = 0 , d 1 = a = 2 η λ − 1 , c 2 = γ − 1. c_1=0,\qquad d_1=\mathfrak{a}=\frac{2\eta}{\lambda}-1,\qquad c_2=\gamma-1. c 1 = 0 , d 1 = a = λ 2 η − 1 , c 2 = γ − 1. Under this matching, the hard-edge constant (3.31) becomes precisely κ γ , λ \kappa_{\gamma,\lambda} κ γ , λ u > 0 u>0 u > 0
lim N → ∞ P ( x min ( N ) > u κ γ , λ 2 N 2 ) = det ( I − K a B e s s e l ) L 2 ( 0 , u ) . (95) \lim_{N\to\infty}\mathbb{P}\left(x_{\min}^{(N)}>\frac{u}{\kappa_{\gamma,\lambda}^{2}N^2}\right)=\det\left(I-K_{\mathfrak{a}}^{\mathrm{Bessel}}\right)_{L^2(0,u)}. \tag*{(95)} N → ∞ lim P ( x m i n ( N ) > κ γ , λ 2 N 2 u ) = det ( I − K a Bessel ) L 2 ( 0 , u ) . (95) The remainder of the argument is analogous to the soft-edge case above, with (4.29) replacing (4.26), which completes the proof.
Finally, we prove Theorem 2.3. In contrast to Theorems 2.1 and 2.2, no q q q
Proof of Theorem 2.3. Write ν : = 2 β + 1 = 2 η / λ − 1 > − 1 \nu:=2\beta+1=2\eta/\lambda-1>-1 ν := 2 β + 1 = 2 η / λ − 1 > − 1 ℓ ≥ 0 \ell\geq0 ℓ ≥ 0
P ( H ⌊ γ N ⌋ , N ≤ ℓ ) = P ( y min ( N ) ( ν , ⌊ γ N ⌋ − N ) ≥ e − λ ℓ ) . (96) \mathbb{P}(H_{\lfloor\gamma N\rfloor,N}\leq\ell)=\mathbb{P}\left(y_{\min}^{(N)}(\nu,\lfloor\gamma N\rfloor-N)\geq e^{-\lambda\ell}\right). \tag*{(96)} P ( H ⌊ γ N ⌋ , N ≤ ℓ ) = P ( y m i n ( N ) ( ν , ⌊ γ N ⌋ − N ) ≥ e − λ ℓ ) . (96) where y min ( N ) ( ν , ⌊ γ N ⌋ − N ) y_{\min}^{(N)}(\nu,\lfloor\gamma N\rfloor-N) y m i n ( N ) ( ν , ⌊ γ N ⌋ − N ) N N N ν \nu ν ⌊ γ N ⌋ − N \lfloor\gamma N\rfloor-N ⌊ γ N ⌋ − N
We recall the hard-edge scaling for this JUE. The limiting Wachter density with left parameter of order one and right parameter ( γ − 1 ) N + O ( 1 ) (\gamma-1)N+O(1) ( γ − 1 ) N + O ( 1 )
ρ J ( x ) = γ π x ( 1 + o ( 1 ) ) , x ↓ 0. \rho_J(x)=\frac{\sqrt{\gamma}}{\pi\sqrt{x}}\left(1+o(1)\right),\qquad x\downarrow0. ρ J ( x ) = π x γ ( 1 + o ( 1 ) ) , x ↓ 0. Then the standard hard-edge scaling theorem for the JUE (see e.g. [60 Theorem 9 and Section 4.2] ) gives, for every u > 0 u>0 u > 0
lim N → ∞ P ( 4 γ N 2 y min ( N ) ( ν , ⌊ γ N ⌋ − N ) ≥ u ) = det ( I − K ν B e s s e l ) L 2 ( 0 , u ) . (97) \lim_{N\to\infty}\mathbb{P}\left(4\gamma N^2y_{\min}^{(N)}(\nu,\lfloor\gamma N\rfloor-N)\geq u\right)=\det\left(I-K_{\nu}^{\mathrm{Bessel}}\right)_{L^2(0,u)}. \tag*{(97)} N → ∞ lim P ( 4 γ N 2 y m i n ( N ) ( ν , ⌊ γ N ⌋ − N ) ≥ u ) = det ( I − K ν Bessel ) L 2 ( 0 , u ) . (97) By (96) , we have
P [ H ⌊ γ N ⌋ , N − 2 λ log N ≤ ξ + log γ λ ] = P ( γ N 2 y min ( N ) ( ν , ⌊ γ N ⌋ − N ) ≥ e − ξ ) = P ( 4 γ N 2 y min ( N ) ( ν , ⌊ γ N ⌋ − N ) ≥ 4 e − ξ ) . (98) \begin{aligned}
\mathbb{P}\left[H_{\lfloor\gamma N\rfloor,N}-\frac{2}{\lambda}\log N\leq\frac{\xi+\log\gamma}{\lambda}\right]=\mathbb{P}\left(\gamma N^2y_{\min}^{(N)}(\nu,\lfloor\gamma N\rfloor-N)\geq e^{-\xi}\right) \\
&=\mathbb{P}\left(4\gamma N^2y_{\min}^{(N)}(\nu,\lfloor\gamma N\rfloor-N)\geq4e^{-\xi}\right). \tag*{(98)}
\end{aligned} P [ H ⌊ γ N ⌋ , N − λ 2 log N ≤ λ ξ + log γ ] = P ( γ N 2 y m i n ( N ) ( ν , ⌊ γ N ⌋ − N ) ≥ e − ξ ) = P ( 4 γ N 2 y m i n ( N ) ( ν , ⌊ γ N ⌋ − N ) ≥ 4 e − ξ ) . (98) Applying (97) with u = 4 e − ξ u=4e^{-\xi} u = 4 e − ξ
lim N → ∞ P [ H ⌊ γ N ⌋ , N − 2 λ log N ≤ ξ + log γ λ ] = det ( I − K ν B e s s e l ) L 2 ( 0 , 4 e − ξ ) . (99) \lim_{N\to\infty}\mathbb{P}\left[H_{\lfloor\gamma N\rfloor,N}-\frac{2}{\lambda}\log N\leq\frac{\xi+\log\gamma}{\lambda}\right]=\det\left(I-K_{\nu}^{\mathrm{Bessel}}\right)_{L^2(0,4e^{-\xi})}. \tag*{(99)} N → ∞ lim P [ H ⌊ γ N ⌋ , N − λ 2 log N ≤ λ ξ + log γ ] = det ( I − K ν Bessel ) L 2 ( 0 , 4 e − ξ ) . (99) It remains only to rewrite the Fredholm determinant in the form stated in the theorem. By the Fredholm expansion,
det ( I − K ν B e s s e l ) L 2 ( 0 , 4 e − ξ ) = 1 + ∑ k = 1 ∞ ( − 1 ) k k ! ∫ ( 0 , 4 e − ξ ) k det ( K ν B e s s e l ( x i , x j ) ) i , j = 1 k ∏ j = 1 k d x j . \det\left(I-K_{\nu}^{\mathrm{Bessel}}\right)_{L^2(0,4e^{-\xi})}=1+\sum_{k=1}^{\infty}\frac{(-1)^k}{k!}\int_{(0,4e^{-\xi})^k}\det\left(K_{\nu}^{\mathrm{Bessel}}(x_i,x_j)\right)_{i,j=1}^k\prod_{j=1}^k dx_j. det ( I − K ν Bessel ) L 2 ( 0 , 4 e − ξ ) = 1 + k = 1 ∑ ∞ k ! ( − 1 ) k ∫ ( 0 , 4 e − ξ ) k det ( K ν Bessel ( x i , x j ) ) i , j = 1 k j = 1 ∏ k d x j . In each integral, make the change of variables x j = 4 e − u j x_j=4e^{-u_j} x j = 4 e − u j j = 1 , … , k j=1,\ldots,k j = 1 , … , k
∫ ( 0 , 4 e − ξ ) k det ( K ν B e s s e l ( x i , x j ) ) i , j = 1 k ∏ j = 1 k d x j = ∫ ( ξ , ∞ ) k det ( K ν B e s s e l ( 4 e − u i , 4 e − u j ) ) i , j = 1 k ∏ j = 1 k 4 e − u j d u j = ∫ ( ξ , ∞ ) k det ( 4 e − ( u i + u j ) / 2 K ν B e s s e l ( 4 e − u i , 4 e − u j ) ) i , j = 1 k ∏ j = 1 k d u j . \begin{aligned}
\int_{(0,4e^{-\xi})^k}\det\left(K_{\nu}^{\mathrm{Bessel}}(x_i,x_j)\right)_{i,j=1}^k\prod_{j=1}^k dx_j
=
\int_{(\xi,\infty)^k}\det\left(K_{\nu}^{\mathrm{Bessel}}(4e^{-u_i},4e^{-u_j})\right)_{i,j=1}^k\prod_{j=1}^k4e^{-u_j}\,du_j \\
&=\int_{(\xi,\infty)^k}\det\left(4e^{-(u_i+u_j)/2}K_{\nu}^{\mathrm{Bessel}}(4e^{-u_i},4e^{-u_j})\right)_{i,j=1}^k\prod_{j=1}^kdu_j.
\end{aligned} ∫ ( 0 , 4 e − ξ ) k det ( K ν Bessel ( x i , x j ) ) i , j = 1 k j = 1 ∏ k d x j = ∫ ( ξ , ∞ ) k det ( K ν Bessel ( 4 e − u i , 4 e − u j ) ) i , j = 1 k j = 1 ∏ k 4 e − u j d u j = ∫ ( ξ , ∞ ) k det ( 4 e − ( u i + u j ) /2 K ν Bessel ( 4 e − u i , 4 e − u j ) ) i , j = 1 k j = 1 ∏ k d u j . Since ν = 2 β + 1 \nu=2\beta+1 ν = 2 β + 1 (30) follows from (99) . Finally, as in the geometric case above, the law of large numbers (29) follows immediately from the fluctuation limit, since the centred fluctuations are of order O ( 1 ) O(1) O ( 1 ) ( 2 / λ ) log N (2/\lambda)\log N ( 2/ λ ) log N
Equilibrium measure and proof of Theorem 3.1
This section is devoted to the proof of Theorem 3.1 . Our approach follows the strategy developed in [31 ] for the q q q [64 ] of the limiting density (55) for the Meixner ensemble, although the form of the constraint is different due to the underlying lattice structure.
More precisely, the large-N N N q q q q q q Theorem 3.1 satisfies these conditions, which, by uniqueness of the equilibrium measure, completes the proof.
While this verification provides a complete proof of the theorem, it does not explain how the explicit formula for the limiting density is found in the first place. For completeness, and to make the origin of this formula transparent, we present its derivation in Appendix A based on the moment method.
We now introduce the equilibrium measure problem. Recall that the weight function w ( a , b ) w^{(a,b)} w ( a , b ) (35) . Under the scaling (43) , the asymptotic formula for the q q q [77 Theorem 4] yields
log w ( a , b ) ( x ; q ) = − N V ( x ) + O ( 1 ) , N → ∞ , (100) \log w^{(a,b)}(x;q)=-NV(x)+O(1),\qquad N\to\infty, \tag*{(100)} log w ( a , b ) ( x ; q ) = − N V ( x ) + O ( 1 ) , N → ∞ , (100) locally uniformly for x ∈ ( 0 , 1 ) x\in(0,1) x ∈ ( 0 , 1 )
V ( x ) : = 1 λ ( Li 2 ( x ) − Li 2 ( s c 2 x ) ) − c 1 log x , x ∈ ( 0 , 1 ) . (101) V(x):=\frac{1}{\lambda}\left(\operatorname{Li}_2(x)-\operatorname{Li}_2(s^{c_2}x)\right)-c_1\log x,\qquad x\in(0,1). \tag*{(101)} V ( x ) := λ 1 ( Li 2 ( x ) − Li 2 ( s c 2 x ) ) − c 1 log x , x ∈ ( 0 , 1 ) . (101) Here, Li 2 \operatorname{Li}_2 Li 2
Li 2 ( x ) : = − ∫ 0 x log ( 1 − t ) t d t . (102) \operatorname{Li}_2(x):=-\int_0^x\frac{\log(1-t)}{t}\,dt. \tag*{(102)} Li 2 ( x ) := − ∫ 0 x t log ( 1 − t ) d t . (102) The associated energy functional is
I [ μ ] = − ∫ 0 1 ∫ 0 1 log ∣ x − y ∣ μ ( d x ) μ ( d y ) + ∫ 0 1 V ( x ) μ ( d x ) (103) I[\mu] = -\int_0^1 \int_0^1 \log|x-y|\,\mu(dx)\,\mu(dy) + \int_0^1 V(x)\,\mu(dx) \tag*{(103)} I [ μ ] = − ∫ 0 1 ∫ 0 1 log ∣ x − y ∣ μ ( d x ) μ ( d y ) + ∫ 0 1 V ( x ) μ ( d x ) (103) over the class of probability measures satisfying the upper constraint
P λ ( 0 , 1 ) : = { μ ( d x ) | supp μ ⊂ [ 0 , 1 ] , d μ d x ≤ 1 λ x } . (104) \mathcal{P}_{\lambda}(0,1) := \left\{\mu(dx)\,\middle|\,\operatorname{supp}\mu\subset[0,1],\ \frac{d\mu}{dx} \le\frac{1}{\lambda x}\right\}. \tag*{(104)} P λ ( 0 , 1 ) := { μ ( d x ) supp μ ⊂ [ 0 , 1 ] , d x d μ ≤ λ x 1 } . (104) Here, the upper constraint reflects the spacing of the underlying q q q [31 Remark 7] ; see also [95 ] for its appearance in the study of limiting zero distributions of orthogonal polynomials.
Following the large-deviation argument of [61 Theorem 1.2] with η = θ = 1 \eta=\theta=1 η = θ = 1 [31 Theorem 1.2(B)] , the empirical measures concentrate exponentially at speed N 2 N^2 N 2 I I I P λ ( 0 , 1 ) . \mathcal{P}_{\lambda}(0,1). P λ ( 0 , 1 ) . d 1 , d 2 d_1,d_2 d 1 , d 2 ρ ( c 1 , c 2 ) ( x ) d x \rho^{(c_1,c_2)}(x)\,dx ρ ( c 1 , c 2 ) ( x ) d x
A standard way to identify the equilibrium measure is to verify the associated Euler–Lagrange variational conditions (see e.g. [8 Section 2.1.4] ). Accordingly, the proof of Theorem 3.1 reduces to the following proposition.
Proposition 5.1. Let μ ( d x ) = ρ ( c 1 , c 2 ) ( x ) d x \mu(dx)=\rho^{(c_1,c_2)}(x)\,dx μ ( d x ) = ρ ( c 1 , c 2 ) ( x ) d x ρ ( c 1 , c 2 ) \rho^{(c_1,c_2)} ρ ( c 1 , c 2 )
d : = { b if λ ≤ λ c , 1 if λ > λ c . (105) d := \begin{cases}
b & \text{if }\lambda\le\lambda_c,\\
1 & \text{if }\lambda> \lambda_c.
\end{cases} \tag*{(105)} d := { b 1 if λ ≤ λ c , if λ > λ c . (105) Then there exists a constant Ω ∈ R \Omega\in\mathbb{R} Ω ∈ R
− 2 ∫ a d log ∣ x − y ∣ μ ( d y ) + V ( x ) { = − Ω if x ∈ ( a , b ) , ≥ − Ω if x ∈ ( 0 , a ) ∪ ( d , 1 ) , ≤ − Ω if x ∈ ( b , d ) . (106) -2\int_a^d \log|x-y|\,\mu(dy)+V(x)
\begin{cases}
=-\Omega& \text{if }x\in(a,b),\\
\ge-\Omega& \text{if }x\in(0,a)\cup(d,1),\\
\le-\Omega& \text{if }x\in(b,d).
\end{cases} \tag*{(106)} − 2 ∫ a d log ∣ x − y ∣ μ ( d y ) + V ( x ) ⎩ ⎨ ⎧ = − Ω ≥ − Ω ≤ − Ω if x ∈ ( a , b ) , if x ∈ ( 0 , a ) ∪ ( d , 1 ) , if x ∈ ( b , d ) . (106) Before the proof of Proposition 5.1, we first evaluate the Cauchy transform
g ( z ) : = ∫ a d μ ( d x ) z − x . (107) g(z):=\int_a^d \frac{\mu(dx)}{z-x}. \tag*{(107)} g ( z ) := ∫ a d z − x μ ( d x ) . (107) For this, recalling that a a a b b b
R ( z ) : = ( B z − 1 − s c 1 ) 2 − 4 s c 1 ( 1 − z ) ( 1 − s c 2 z ) = ( s − 1 − s c 1 + c 2 + 1 ) ( z − a ) ( z − b ) , (108) R(z):=\sqrt{(Bz-1-s^{c_1})^2-4s^{c_1}(1-z)(1-s^{c_2}z)}
=(s^{-1}-s^{c_1+c_2+1})\sqrt{(z-a)(z-b)}, \tag*{(108)} R ( z ) := ( B z − 1 − s c 1 ) 2 − 4 s c 1 ( 1 − z ) ( 1 − s c 2 z ) = ( s − 1 − s c 1 + c 2 + 1 ) ( z − a ) ( z − b ) , (108) where
B : = s − 1 + s c 1 + c 2 + 1 . (109) B:=s^{-1}+s^{c_1+c_2+1}. \tag*{(109)} B := s − 1 + s c 1 + c 2 + 1 . (109) Here, the branch is analytic in C ∖ [ a , b ] \mathbb{C}\setminus[a,b] C ∖ [ a , b ] R ( z ) ∼ ( s − 1 − s c 1 + c 2 + 1 ) z R(z)\sim(s^{-1}-s^{c_1+c_2+1})z R ( z ) ∼ ( s − 1 − s c 1 + c 2 + 1 ) z z → ∞ z\to\infty z → ∞
Lemma 5.2. We have
g ( z ) = 1 λ z log B z − 1 − s c 1 + R ( z ) 2 ( z − 1 ) , z ∈ C ∖ [ a , d ] . (110) g(z)=\frac{1}{\lambda z}\log\frac{Bz-1-s^{c_1}+R(z)}{2(z-1)},\qquad z\in\mathbb{C}\setminus[a,d]. \tag*{(110)} g ( z ) = λ z 1 log 2 ( z − 1 ) B z − 1 − s c 1 + R ( z ) , z ∈ C ∖ [ a , d ] . (110) The logarithm is chosen by analytic continuation from z > 1 z>1 z > 1
g ± ( x ) = 1 2 λ x log s c 1 ( 1 − s c 2 x ) 1 − x ∓ π i ρ ( c 1 , c 2 ) ( x ) , x ∈ ( a , b ) , (111) g_{\pm}(x)=\frac{1}{2\lambda x}\log\frac{s^{c_1}(1-s^{c_2}x)}{1-x}\mp\pi i\rho^{(c_1,c_2)}(x),\qquad x\in(a,b), \tag*{(111)} g ± ( x ) = 2 λ x 1 log 1 − x s c 1 ( 1 − s c 2 x ) ∓ π i ρ ( c 1 , c 2 ) ( x ) , x ∈ ( a , b ) , (111) where g ± g_{\pm} g ±
Proof. By (5.9), our choice of branch implies that R ( x ) < 0 R(x)<0 R ( x ) < 0 x < a x<a x < a R ( x ) > 0 R(x)>0 R ( x ) > 0 x > b x>b x > b
R + ( x ) = − R − ( x ) = i 4 s c 1 ( 1 − x ) ( 1 − s c 2 x ) − ( B x − 1 − s c 1 ) 2 , x ∈ ( a , b ) . (112) R_+(x)=-R_-(x)=i\sqrt{4s^{c_1}(1-x)(1-s^{c_2}x)-(Bx-1-s^{c_1})^2},\qquad x\in(a,b). \tag*{(112)} R + ( x ) = − R − ( x ) = i 4 s c 1 ( 1 − x ) ( 1 − s c 2 x ) − ( B x − 1 − s c 1 ) 2 , x ∈ ( a , b ) . (112) To establish (5.11), it suffices to show that
∫ a d μ ( d x ) z − x = log H ( z ) λ z , H ( z ) : = B z − 1 − s c 1 + R ( z ) 2 ( z − 1 ) . (113) \int_a^d \frac{\mu(dx)}{z-x}=\frac{\log H(z)}{\lambda z},\qquad H(z):=\frac{Bz-1-s^{c_1}+R(z)}{2(z-1)}. \tag*{(113)} ∫ a d z − x μ ( d x ) = λ z log H ( z ) , H ( z ) := 2 ( z − 1 ) B z − 1 − s c 1 + R ( z ) . (113) We verify this identity by computing the boundary jump of log H ( z ) / ( λ z ) \log H(z)/(\lambda z) log H ( z ) / ( λ z ) [ a , d ] [a,d] [ a , d ] log H ( z ) / ( λ z ) \log H(z)/(\lambda z) log H ( z ) / ( λ z ) (108) ,
( B z − 1 − s c 1 + R ( z ) ) ( B z − 1 − s c 1 − R ( z ) ) = 4 s c 1 ( 1 − z ) ( 1 − s c 2 z ) . (114) (Bz-1-s^{c_1}+R(z))(Bz-1-s^{c_1}-R(z))=4s^{c_1}(1-z)(1-s^{c_2}z). \tag*{(114)} ( B z − 1 − s c 1 + R ( z )) ( B z − 1 − s c 1 − R ( z )) = 4 s c 1 ( 1 − z ) ( 1 − s c 2 z ) . (114) Consequently, a zero of the numerator of H H H z = 1 z=1 z = 1 z = s − c 2 z=s^{-c_2} z = s − c 2 c 2 > 0 c_2>0 c 2 > 0
B s − c 2 − 1 − s c 1 = ( 1 − s ) ( s − 1 − s c 1 ) + s − 1 ( s − c 2 − 1 ) > 0. Bs^{-c_2}-1-s^{c_1}=(1-s)(s^{-1}-s^{c_1})+s^{-1}(s^{-c_2}-1)>0. B s − c 2 − 1 − s c 1 = ( 1 − s ) ( s − 1 − s c 1 ) + s − 1 ( s − c 2 − 1 ) > 0. Since c 2 > 0 c_2>0 c 2 > 0 s − c 2 > 1 ≥ b s^{-c_2}>1\ge b s − c 2 > 1 ≥ b R ( s − c 2 ) > 0 R(s^{-c_2})>0 R ( s − c 2 ) > 0 z = 1 z=1 z = 1
B { < 1 + s c 1 , if λ < λ c , = 1 + s c 1 , if λ = λ c , > 1 + s c 1 , if λ > λ c . (115) B\begin{cases}
<1+s^{c_1}, & \text{if }\lambda<\lambda_c,\\
=1+s^{c_1}, & \text{if }\lambda=\lambda_c,\\
>1+s^{c_1}, & \text{if }\lambda>\lambda_c.
\end{cases} \tag*{(115)} B ⎩ ⎨ ⎧ < 1 + s c 1 , = 1 + s c 1 , > 1 + s c 1 , if λ < λ c , if λ = λ c , if λ > λ c . (115) If λ < λ c \lambda<\lambda_c λ < λ c R ( 1 ) = 1 + s c 1 − B R(1)=1+s^{c_1}-B R ( 1 ) = 1 + s c 1 − B (113) and (114) , we have
H ( z ) = 2 s c 1 ( 1 − s c 2 z ) 1 + s c 1 − B z + R ( z ) . H(z)=\frac{2s^{c_1}(1-s^{c_2}z)}{1+s^{c_1}-Bz+R(z)}. H ( z ) = 1 + s c 1 − B z + R ( z ) 2 s c 1 ( 1 − s c 2 z ) . Therefore, the singularity at z = 1 z=1 z = 1 H ( 1 ) > 0 H(1)>0 H ( 1 ) > 0 1 ∈ [ a , d ] 1\in[a,d] 1 ∈ [ a , d ] H H H C ∖ [ a , d ] \mathbb{C}\setminus[a,d] C ∖ [ a , d ]
On the other hand, the normalisation R ( z ) ∼ ( s − 1 − s c 1 + c 2 + 1 ) z R(z)\sim(s^{-1}-s^{c_1+c_2+1})z R ( z ) ∼ ( s − 1 − s c 1 + c 2 + 1 ) z z → ∞ z\to\infty z → ∞
H ( z ) = s − 1 + O ( z − 1 ) , log H ( z ) = λ + O ( z − 1 ) , z → ∞ . (116) H(z)=s^{-1}+O(z^{-1}),\qquad\log H(z)=\lambda+O(z^{-1}),\qquad z\to\infty. \tag*{(116)} H ( z ) = s − 1 + O ( z − 1 ) , log H ( z ) = λ + O ( z − 1 ) , z → ∞. (116) If a > 0 a>0 a > 0 c 1 > 0 c_1>0 c 1 > 0
R ( 0 ) = − ( 1 − s c 1 ) , H ( 0 ) = 1. R(0)=-(1-s^{c_1}),\qquad H(0)=1. R ( 0 ) = − ( 1 − s c 1 ) , H ( 0 ) = 1. Moreover, H ( x ) > 0 H(x)>0 H ( x ) > 0 x < 0 x<0 x < 0 log H ( 0 ) = 0 \log H(0)=0 log H ( 0 ) = 0 log H ( z ) / z \log H(z)/z log H ( z ) / z z = 0 z=0 z = 0
We next determine the boundary values of log H ( z ) \log H(z) log H ( z ) λ < λ c \lambda<\lambda_c λ < λ c H H H ( b , ∞ ) (b,\infty) ( b , ∞ ) s − 1 > 0 s^{-1}>0 s − 1 > 0 λ ≥ λ c \lambda\ge\lambda_c λ ≥ λ c z → 1 z\to1 z → 1
H ( z ) = { s c 1 ( 1 − s c 2 ) z − 1 + O ( 1 ) , if λ = λ c , B − 1 − s c 1 z − 1 + O ( 1 ) , if λ > λ c . (117) H(z)=\begin{cases}
\frac{\sqrt{s^{c_1}(1-s^{c_2})}}{\sqrt{z-1}}+O(1), & \text{if }\lambda=\lambda_c,\\[6pt]
\frac{B-1-s^{c_1}}{z-1}+O(1), & \text{if }\lambda>\lambda_c.
\end{cases} \tag*{(117)} H ( z ) = ⎩ ⎨ ⎧ z − 1 s c 1 ( 1 − s c 2 ) + O ( 1 ) , z − 1 B − 1 − s c 1 + O ( 1 ) , if λ = λ c , if λ > λ c . (117) When λ = λ c \lambda=\lambda_c λ = λ c b = 1 b=1 b = 1 arg H + ( x ) → − π / 2 \arg H_+(x)\to-\pi/2 arg H + ( x ) → − π /2 x ↑ 1 x\uparrow1 x ↑ 1 λ > λ c \lambda>\lambda_c λ > λ c (117) has positive residue. Since H H H ( b , 1 ) (b,1) ( b , 1 ) z > 1 z>1 z > 1
log H ± ( x ) = log ∣ H ( x ) ∣ ∓ π i , x ∈ ( b , d ) . (118) \log H_{\pm}(x)=\log|H(x)|\mp\pi i,\qquad x\in(b,d). \tag*{(118)} log H ± ( x ) = log ∣ H ( x ) ∣ ∓ π i , x ∈ ( b , d ) . (118) For x ∈ ( a , b ) x\in(a,b) x ∈ ( a , b ) (112) , we have
H + ( x ) = 1 + s c 1 − B x − R + ( x ) 2 ( 1 − x ) , H − ( x ) = H + ( x ) ‾ . H_+(x)=\frac{1+s^{c_1}-Bx-R_+(x)}{2(1-x)},\qquad H_-(x)=\overline{H_+(x)}. H + ( x ) = 2 ( 1 − x ) 1 + s c 1 − B x − R + ( x ) , H − ( x ) = H + ( x ) . In particular, Im H + ( x ) < 0 \operatorname{Im}H_+(x)<0 Im H + ( x ) < 0
∣ H + ( x ) ∣ 2 = ( B x − 1 − s c 1 ) 2 + ∣ R + ( x ) ∣ 2 4 ( 1 − x ) 2 = s c 1 ( 1 − s c 2 x ) 1 − x . |H_+(x)|^2=\frac{(Bx-1-s^{c_1})^2+|R_+(x)|^2}{4(1-x)^2}=\frac{s^{c_1}(1-s^{c_2}x)}{1-x}. ∣ H + ( x ) ∣ 2 = 4 ( 1 − x ) 2 ( B x − 1 − s c 1 ) 2 + ∣ R + ( x ) ∣ 2 = 1 − x s c 1 ( 1 − s c 2 x ) . The branch determination above shows that the argument of H + ( x ) H_+(x) H + ( x ) ( − π , 0 ) (-\pi,0) ( − π , 0 ) ( a , b ) (a,b) ( a , b )
arg H + ( x ) = − arccos 1 + s c 1 − B x 2 s c 1 ( 1 − x ) ( 1 − s c 2 x ) = − π λ x ρ ( c 1 , c 2 ) ( x ) , \arg H_+(x)=-\arccos\frac{1+s^{c_1}-Bx}{2\sqrt{s^{c_1}(1-x)(1-s^{c_2}x)}}=-\pi\lambda x\rho^{(c_1,c_2)}(x), arg H + ( x ) = − arccos 2 s c 1 ( 1 − x ) ( 1 − s c 2 x ) 1 + s c 1 − B x = − π λ x ρ ( c 1 , c 2 ) ( x ) , where the last equality follows from (48) . Combining the above, we obtain
log H ± ( x ) = 1 2 log s c 1 ( 1 − s c 2 x ) 1 − x ∓ π i λ x ρ ( c 1 , c 2 ) ( x ) , x ∈ ( a , b ) . (119) \log H_{\pm}(x)=\frac{1}{2}\log\frac{s^{c_1}(1-s^{c_2}x)}{1-x}\mp\pi i\lambda x\rho^{(c_1,c_2)}(x),\qquad x\in(a,b). \tag*{(119)} log H ± ( x ) = 2 1 log 1 − x s c 1 ( 1 − s c 2 x ) ∓ π iλ x ρ ( c 1 , c 2 ) ( x ) , x ∈ ( a , b ) . (119) Together with (118) and the identity ρ ( c 1 , c 2 ) ( x ) = 1 / ( λ x ) \rho^{(c_1,c_2)}(x)=1/(\lambda x) ρ ( c 1 , c 2 ) ( x ) = 1/ ( λ x ) ( b , d ) (b,d) ( b , d )
log H + ( x ) − log H − ( x ) λ x = − 2 π i ρ ( c 1 , c 2 ) ( x ) , x ∈ ( a , d ) ∖ { b } . (120) \frac{\log H_{+}(x)-\log H_{-}(x)}{\lambda x}=-2\pi i\rho^{(c_1,c_2)}(x),\qquad x\in(a,d)\setminus\{b\}. \tag*{(120)} λ x log H + ( x ) − log H − ( x ) = − 2 π i ρ ( c 1 , c 2 ) ( x ) , x ∈ ( a , d ) ∖ { b } . (120) Thus, applying the Sokhotski–Plemelj formula together with (116) and (120) , we obtain
log H ( z ) λ z = 1 2 π i ∫ a d log H + ( x ) − log H − ( x ) λ x ( x − z ) d x = ∫ a d ρ ( c 1 , c 2 ) ( x ) z − x d x . \frac{\log H(z)}{\lambda z}=\frac{1}{2\pi i}\int_a^d\frac{\log H_{+}(x)-\log H_{-}(x)}{\lambda x(x-z)}\,dx=\int_a^d\frac{\rho^{(c_1,c_2)}(x)}{z-x}\,dx. λ z log H ( z ) = 2 π i 1 ∫ a d λ x ( x − z ) log H + ( x ) − log H − ( x ) d x = ∫ a d z − x ρ ( c 1 , c 2 ) ( x ) d x . This proves (110) , while (111) follows from (119) .
Using Lemma 5.2, we now prove Proposition 5.1.
Proof of Proposition 5.1. We use the notation of Lemma 5.2 and set
E ( x ) : = − 2 ∫ a d log ∣ x − y ∣ μ ( d y ) + V ( x ) . (121) \mathcal{E}(x):=-2\int_a^d\log|x-y|\,\mu(dy)+V(x). \tag*{(121)} E ( x ) := − 2 ∫ a d log ∣ x − y ∣ μ ( d y ) + V ( x ) . (121) By using log s = − λ \log s=-\lambda log s = − λ (102) , we have
V ′ ( x ) = 1 λ x log s c 1 ( 1 − s c 2 x ) 1 − x . (122) V'(x)=\frac{1}{\lambda x}\log\frac{s^{c_1}(1-s^{c_2}x)}{1-x}. \tag*{(122)} V ′ ( x ) = λ x 1 log 1 − x s c 1 ( 1 − s c 2 x ) . (122) Thus, the boundary values (111) satisfy
g + ( x ) + g − ( x ) = V ′ ( x ) , x ∈ ( a , b ) . g_{+}(x)+g_{-}(x)=V'(x),\qquad x\in(a,b). g + ( x ) + g − ( x ) = V ′ ( x ) , x ∈ ( a , b ) . By the Sokhotski–Plemelj formulas, it follows that for x ∈ ( a , b ) x\in(a,b) x ∈ ( a , b )
E ′ ( x ) = V ′ ( x ) − 2 p . v . ∫ a d μ ( d y ) x − y = V ′ ( x ) − g + ( x ) − g − ( x ) = 0. \mathcal{E}'(x)=V'(x)-2\mathop{\mathrm{p.v.}}\int_a^d\frac{\mu(dy)}{x-y}=V'(x)-g_{+}(x)-g_{-}(x)=0. E ′ ( x ) = V ′ ( x ) − 2 p.v. ∫ a d x − y μ ( d y ) = V ′ ( x ) − g + ( x ) − g − ( x ) = 0. Consequently, there exists a constant Ω ∈ R \Omega\in\mathbb{R} Ω ∈ R E ( x ) = − Ω \mathcal{E}(x)=-\Omega E ( x ) = − Ω x ∈ ( a , b ) x\in(a,b) x ∈ ( a , b ) (106) .
We next verify the inequalities in (106) . For x ∈ ( 0 , a ) ∪ ( b , 1 ) x\in(0,a)\cup(b,1) x ∈ ( 0 , a ) ∪ ( b , 1 ) R ( x ) R(x) R ( x ) (108) , (110) and (122) , we obtain
E ′ ( x ) = V ′ ( x ) − 2 Re g + ( x ) = 1 λ x log 4 s c 1 ( 1 − x ) ( 1 − s c 2 x ) ( B x − 1 − s c 1 + R ( x ) ) 2 = 1 λ x log B x − 1 − s c 1 − R ( x ) B x − 1 − s c 1 + R ( x ) . (123) \mathcal{E}'(x)=V'(x)-2\operatorname{Re}g_{+}(x)=\frac{1}{\lambda x}\log\frac{4s^{c_1}(1-x)(1-s^{c_2}x)}{(Bx-1-s^{c_1}+R(x))^2}=\frac{1}{\lambda x}\log\frac{Bx-1-s^{c_1}-R(x)}{Bx-1-s^{c_1}+R(x)}. \tag*{(123)} E ′ ( x ) = V ′ ( x ) − 2 Re g + ( x ) = λ x 1 log ( B x − 1 − s c 1 + R ( x ) ) 2 4 s c 1 ( 1 − x ) ( 1 − s c 2 x ) = λ x 1 log B x − 1 − s c 1 + R ( x ) B x − 1 − s c 1 − R ( x ) . (123) Moreover, by using s c 1 ( 1 − x ) ( 1 − s c 2 x ) > 0 s^{c_1}(1-x)(1-s^{c_2}x)>0 s c 1 ( 1 − x ) ( 1 − s c 2 x ) > 0 x ∈ ( 0 , 1 ) x\in(0,1) x ∈ ( 0 , 1 ) (108) , we have
0 < ∣ R ( x ) ∣ < ∣ B x − 1 − s c 1 ∣ , x ∈ ( 0 , a ) ∪ ( b , 1 ) . (124) 0<|R(x)|<|Bx-1-s^{c_1}|,\qquad x\in(0,a)\cup(b,1). \tag*{(124)} 0 < ∣ R ( x ) ∣ < ∣ B x − 1 − s c 1 ∣ , x ∈ ( 0 , a ) ∪ ( b , 1 ) . (124) In particular, the argument of the logarithm in (123) is positive.
Now it remains to determine the sign of E ′ \mathcal{E}' E ′ λ ≤ λ c \lambda\leq\lambda_c λ ≤ λ c (115) , we have B x − 1 − s c 1 < 0 Bx-1-s^{c_1}<0 B x − 1 − s c 1 < 0 x ∈ ( 0 , 1 ) x\in(0,1) x ∈ ( 0 , 1 ) λ > λ c \lambda>\lambda_c λ > λ c ( 1 + s c 1 ) / B (1+s^{c_1})/B ( 1 + s c 1 ) / B ( 0 , 1 ) (0,1) ( 0 , 1 ) (108) is negative at this point. Then by (108) , we have
a < 1 + s c 1 B < b . a<\frac{1+s^{c_1}}{B}<b. a < B 1 + s c 1 < b . Together with the definition (105) of d d d
B x − 1 − s c 1 { < 0 , if x ∈ ( 0 , a ) ∪ ( d , 1 ) , > 0 , if x ∈ ( b , d ) . Bx-1-s^{c_1}
\begin{cases}
<0, & \text{if }x\in(0,a)\cup(d,1),\\
>0, & \text{if }x\in(b,d).
\end{cases} B x − 1 − s c 1 { < 0 , > 0 , if x ∈ ( 0 , a ) ∪ ( d , 1 ) , if x ∈ ( b , d ) . On ( 0 , a ) (0,a) ( 0 , a ) B x − 1 − s c 1 Bx-1-s^{c_1} B x − 1 − s c 1 R ( x ) R(x) R ( x ) (124) shows that the ratio in (123) lies in ( 0 , 1 ) (0,1) ( 0 , 1 ) ( d , 1 ) (d,1) ( d , 1 ) R ( x ) > 0 R(x)>0 R ( x ) > 0 ( b , d ) (b,d) ( b , d ) ( 0 , 1 ) (0,1) ( 0 , 1 )
E ′ ( x ) { < 0 , if x ∈ ( 0 , a ) ∪ ( b , d ) , > 0 , if x ∈ ( d , 1 ) . \mathcal{E}'(x)
\begin{cases}
<0, & \text{if }x\in(0,a)\cup(b,d),\\
>0, & \text{if }x\in(d,1).
\end{cases} E ′ ( x ) { < 0 , > 0 , if x ∈ ( 0 , a ) ∪ ( b , d ) , if x ∈ ( d , 1 ) . By continuity, E ( a ) = E ( b ) = − Ω \mathcal{E}(a)=\mathcal{E}(b)=-\Omega E ( a ) = E ( b ) = − Ω (106) now follow. □
Fluctuations and proof of Theorem 3.2
In this section, we prove Theorem 3.2. When c 2 > 0 c_{2}>0 c 2 > 0 d 2 d_{2} d 2 R \mathbb{R} R q q q d 2 d_{2} d 2
We first recall the determinantal structure of the q q q q q q q q q { p n ( a , b ) } n ≥ 0 \{p_{n}^{(a,b)}\}_{n\geq0} { p n ( a , b ) } n ≥ 0 q q q
∫ 0 1 p m ( a , b ) ( x ; q ) p n ( a , b ) ( x ; q ) w ( a , b ) ( x ; q ) d q x = h n ( a , b ) δ m n . (125) \int_{0}^{1}p_{m}^{(a,b)}(x;q)p_{n}^{(a,b)}(x;q)w^{(a,b)}(x;q)\,d_{q}x=h_{n}^{(a,b)}\delta_{mn}. \tag*{(125)} ∫ 0 1 p m ( a , b ) ( x ; q ) p n ( a , b ) ( x ; q ) w ( a , b ) ( x ; q ) d q x = h n ( a , b ) δ mn . (125) Here, d q x d_{q}x d q x q q q
h n ( a , b ) = q n ( a + 1 ) ( 1 − q ) 1 − q 2 n + a + b + 1 ( q , q a + b + 1 ; q ) ∞ ( q a + 1 , q b + 1 ; q ) ∞ ( q , q b + 1 ; q ) n ( q a + 1 , q a + b + 1 ; q ) n . (126) h_{n}^{(a,b)}=\frac{q^{n(a+1)}(1-q)}{1-q^{2n+a+b+1}}\frac{(q,q^{a+b+1};q)_{\infty}}{(q^{a+1},q^{b+1};q)_{\infty}}\frac{(q,q^{b+1};q)_{n}}{(q^{a+1},q^{a+b+1};q)_{n}}. \tag*{(126)} h n ( a , b ) = 1 − q 2 n + a + b + 1 q n ( a + 1 ) ( 1 − q ) ( q a + 1 , q b + 1 ; q ) ∞ ( q , q a + b + 1 ; q ) ∞ ( q a + 1 , q a + b + 1 ; q ) n ( q , q b + 1 ; q ) n . (126) is the squared orthogonal norm. (Expressions with removable singularities in the parameters are understood by continuity.) The little q q q
p n ( a , b ) ( x ; q ) = 2 ϕ 1 ( q − n , q n + a + b + 1 q a + 1 ; q , q x ) = ∑ k = 0 n ( q − n , q n + a + b + 1 ; q ) k ( q , q a + 1 ; q ) k ( q x ) k ; (127) p_{n}^{(a,b)}(x;q)={}_{2}\phi_{1}\left(\begin{matrix}q^{-n},q^{n+a+b+1}\\q^{a+1}\end{matrix};q,qx\right)=\sum_{k=0}^{n}\frac{(q^{-n},q^{n+a+b+1};q)_{k}}{(q,q^{a+1};q)_{k}}(qx)^{k}; \tag*{(127)} p n ( a , b ) ( x ; q ) = 2 ϕ 1 ( q − n , q n + a + b + 1 q a + 1 ; q , q x ) = k = 0 ∑ n ( q , q a + 1 ; q ) k ( q − n , q n + a + b + 1 ; q ) k ( q x ) k ; (127) see [71 Eq. (14.12.1)] .
The q q q q q q
K N ( a , b ) ( x , y ; q ) : = w ( a , b ) ( x ; q ) w ( a , b ) ( y ; q ) ∑ j = 0 N − 1 1 h j ( a , b ) p j ( a , b ) ( x ; q ) p j ( a , b ) ( y ; q ) . (128) K_{N}^{(a,b)}(x,y;q):=\sqrt{w^{(a,b)}(x;q)w^{(a,b)}(y;q)}\sum_{j=0}^{N-1}\frac{1}{h_{j}^{(a,b)}}p_{j}^{(a,b)}(x;q)p_{j}^{(a,b)}(y;q). \tag*{(128)} K N ( a , b ) ( x , y ; q ) := w ( a , b ) ( x ; q ) w ( a , b ) ( y ; q ) j = 0 ∑ N − 1 h j ( a , b ) 1 p j ( a , b ) ( x ; q ) p j ( a , b ) ( y ; q ) . (128) The Christoffel-Darboux formula yields the representation of the correlation kernel
K N ( a , b ) ( x , y ; q ) = w ( a , b ) ( x ; q ) w ( a , b ) ( y ; q ) h N − 1 ( a , b ) γ N − 1 ( a , b ) γ N ( a , b ) p N ( a , b ) ( x ; q ) p N − 1 ( a , b ) ( y ; q ) − p N − 1 ( a , b ) ( x ; q ) p N ( a , b ) ( y ; q ) x − y . (129) K_{N}^{(a,b)}(x,y;q)=\frac{\sqrt{w^{(a,b)}(x;q)w^{(a,b)}(y;q)}}{h_{N-1}^{(a,b)}}\frac{\gamma_{N-1}^{(a,b)}}{\gamma_{N}^{(a,b)}}\frac{p_{N}^{(a,b)}(x;q)p_{N-1}^{(a,b)}(y;q)-p_{N-1}^{(a,b)}(x;q)p_{N}^{(a,b)}(y;q)}{x-y}. \tag*{(129)} K N ( a , b ) ( x , y ; q ) = h N − 1 ( a , b ) w ( a , b ) ( x ; q ) w ( a , b ) ( y ; q ) γ N ( a , b ) γ N − 1 ( a , b ) x − y p N ( a , b ) ( x ; q ) p N − 1 ( a , b ) ( y ; q ) − p N − 1 ( a , b ) ( x ; q ) p N ( a , b ) ( y ; q ) . (129) Here,
γ N ( a , b ) = ( − 1 ) N q N ( N − 1 ) 2 ( q N + a + b + 1 ; q ) N ( q a + 1 ; q ) N (130) \gamma_{N}^{(a,b)}=\frac{(-1)^{N}q^{\frac{N(N-1)}{2}}(q^{N+a+b+1};q)_{N}}{(q^{a+1};q)_{N}} \tag*{(130)} γ N ( a , b ) = ( q a + 1 ; q ) N ( − 1 ) N q 2 N ( N − 1 ) ( q N + a + b + 1 ; q ) N (130) is the leading coefficient of the little q q q p N ( a , b ) ( x ) p_{N}^{(a,b)}(x) p N ( a , b ) ( x )
Hard edge scaling limit
Throughout this subsection, we assume c 1 = 0 c_{1}=0 c 1 = 0
Lemma 6.1. Let q = e − λ / N q=e^{-\lambda/N} q = e − λ / N λ > 0 \lambda>0 λ > 0 a = d 1 > − 1 a=d_{1}>-1 a = d 1 > − 1 j ∈ Z ≥ 0 j\in\mathbb{Z}_{\geq0} j ∈ Z ≥ 0 N → ∞ N\to\infty N → ∞
p N − j ( a , b ) ( u ( κ N ) 2 ; q ) = Γ ( d 1 + 1 ) 2 d 1 u − d 1 / 2 J d 1 ( u ) + o ( 1 ) , (131) p_{N-j}^{(a,b)}\left(\frac{u}{(\kappa N)^{2}};q\right)=\Gamma(d_{1}+1)2^{d_{1}}u^{-d_{1}/2}J_{d_{1}}(\sqrt{u})+o(1), \tag*{(131)} p N − j ( a , b ) ( ( κ N ) 2 u ; q ) = Γ ( d 1 + 1 ) 2 d 1 u − d 1 /2 J d 1 ( u ) + o ( 1 ) , (131) locally uniformly for u ∈ C u\in\mathbb{C} u ∈ C u u u
Proof. The basic hypergeometric representation (6.3) gives
p N − j ( a , b ) ( u ( κ N ) 2 ; q ) = ∑ k = 0 N − j ( q − N + j , q N + a + b − j + 1 ; q ) k ( q , q a + 1 ; q ) k q k u k ( κ N ) 2 k . p_{N-j}^{(a,b)}\left(\frac{u}{(\kappa N)^{2}};q\right)=\sum_{k=0}^{N-j}\frac{(q^{-N+j},q^{N+a+b-j+1};q)_{k}}{(q,q^{a+1};q)_{k}}\frac{q^{k}u^{k}}{(\kappa N)^{2k}}. p N − j ( a , b ) ( ( κ N ) 2 u ; q ) = k = 0 ∑ N − j ( q , q a + 1 ; q ) k ( q − N + j , q N + a + b − j + 1 ; q ) k ( κ N ) 2 k q k u k . For each fixed k k k κ \kappa κ
( q − N + j , q N + a + b − j + 1 ; q ) k ( q , q a + 1 ; q ) k q k ( κ N ) 2 k = ( − 1 4 ) k Γ ( d 1 + 1 ) k ! Γ ( d 1 + k + 1 ) + O ( N − 1 ) . (132) \frac{(q^{-N+j},q^{N+a+b-j+1};q)_{k}}{(q,q^{a+1};q)_{k}}\frac{q^{k}}{(\kappa N)^{2k}}=\left(-\frac{1}{4}\right)^{k}\frac{\Gamma(d_{1}+1)}{k!\,\Gamma(d_{1}+k+1)}+O(N^{-1}). \tag*{(132)} ( q , q a + 1 ; q ) k ( q − N + j , q N + a + b − j + 1 ; q ) k ( κ N ) 2 k q k = ( − 4 1 ) k k ! Γ ( d 1 + k + 1 ) Γ ( d 1 + 1 ) + O ( N − 1 ) . (132) To justify passage to the limit in the sum, we derive a summable bound independent of N N N c = min { 1 , d 1 + 1 } > 0 c=\min\{1,d_1+1\}>0 c = min { 1 , d 1 + 1 } > 0 a + ℓ ≥ c ℓ a+\ell\ge c\ell a + ℓ ≥ c ℓ ℓ ≥ 1 \ell\ge1 ℓ ≥ 1
1 − q a + ℓ ≥ 1 − q c ℓ ≥ c ( 1 − q ℓ ) . 1-q^{a+\ell}\ge1-q^{c\ell}\ge c(1-q^\ell). 1 − q a + ℓ ≥ 1 − q c ℓ ≥ c ( 1 − q ℓ ) . Moreover, by concavity of 1 − e − λ t 1-e^{-\lambda t} 1 − e − λ t
1 − q ℓ ≥ ℓ N ( 1 − s ) , 1 ≤ ℓ ≤ N . 1-q^\ell\ge\frac{\ell}{N}(1-s),\qquad1\le\ell\le N. 1 − q ℓ ≥ N ℓ ( 1 − s ) , 1 ≤ ℓ ≤ N . Consequently, for 0 ≤ k ≤ N − j 0\le k\le N-j 0 ≤ k ≤ N − j
( q a + 1 ; q ) k ≥ c k ( q ; q ) k , ( q ; q ) k ≥ ( 1 − s N ) k k ! . (q^{a+1};q)_k\ge c^k(q;q)_k,\qquad(q;q)_k\ge\left(\frac{1-s}{N}\right)^k k!. ( q a + 1 ; q ) k ≥ c k ( q ; q ) k , ( q ; q ) k ≥ ( N 1 − s ) k k ! . For all sufficiently large N N N
∣ ( q − N + j , q N + a + b − j + 1 ; q ) k ∣ ≤ ∣ ( q − N + j ; q ) k ∣ ≤ ( s − 1 − 1 ) k . \left|(q^{-N+j},q^{N+a+b-j+1};q)_k\right|\le\left|(q^{-N+j};q)_k\right|\le(s^{-1}-1)^k. ( q − N + j , q N + a + b − j + 1 ; q ) k ≤ ( q − N + j ; q ) k ≤ ( s − 1 − 1 ) k . Thus, for each fixed R > 0 R>0 R > 0 ∣ u ∣ ≤ R |u|\le R ∣ u ∣ ≤ R
∣ ( q − N + j , q N + a + b − j + 1 ; q ) k ( q , q a + 1 ; q ) k q k u k ( κ N ) 2 k ∣ ≤ 1 ( k ! ) 2 ( R c κ 2 s ( 1 − s ) ) k . \left|\frac{(q^{-N+j},q^{N+a+b-j+1};q)_k}{(q, q^{a+1};q)_k}\frac{q^k u^k}{(\kappa N)^{2k}}\right|
\le\frac{1}{(k!)^2}\left(\frac{R}{c\kappa^2s(1-s)}\right)^k. ( q , q a + 1 ; q ) k ( q − N + j , q N + a + b − j + 1 ; q ) k ( κ N ) 2 k q k u k ≤ ( k ! ) 2 1 ( c κ 2 s ( 1 − s ) R ) k . The right-hand side is summable over k ≥ 0 k\ge0 k ≥ 0 k > N − j k>N-j k > N − j ∣ u ∣ ≤ R |u|\le R ∣ u ∣ ≤ R R R R C \mathbb{C} C
We next use the polynomial asymptotics to derive the hard-edge limit of the correlation kernel. To obtain uniform convergence up to the origin for the full range d 1 > − 1 d_1>-1 d 1 > − 1 ( u v ) d 1 / 2 (uv)^{d_1/2} ( uv ) d 1 /2
Proposition 6.2. Let q = e − λ / N q=e^{-\lambda/N} q = e − λ / N λ > 0 \lambda>0 λ > 0 a = d 1 > − 1 a=d_1>-1 a = d 1 > − 1 b = c 2 N + d 2 b=c_2N+d_2 b = c 2 N + d 2 M > 0 M>0 M > 0
L M h a r d = { u ∈ ( 0 , M ) : u ( κ N ) 2 ∈ L q } , (133) \mathcal{L}^{\mathrm{hard}}_M=\left\{u\in(0,M):\frac{u}{(\kappa N)^2}\in L_q\right\}, \tag*{(133)} L M hard = { u ∈ ( 0 , M ) : ( κ N ) 2 u ∈ L q } , (133) where L q = { 1 , q , q 2 , … } L_q=\{1,q,q^2,\ldots\} L q = { 1 , q , q 2 , … } N → ∞ N\to\infty N → ∞
sup u , v ∈ L M h a r d ( u v ) − d 1 / 2 ∣ 1 ( κ N ) 2 K N ( a , b ) ( u ( κ N ) 2 , v ( κ N ) 2 ; q ) − K d 1 B e s s e l ( u , v ) ∣ → 0. (134) \sup_{u,v\in\mathcal{L}^{\mathrm{hard}}_M}(uv)^{-d_1/2}\left|\frac{1}{(\kappa N)^2}K_N^{(a,b)}\left(\frac{u}{(\kappa N)^2},\frac{v}{(\kappa N)^2};q\right)-K_{d_1}^{\mathrm{Bessel}}(u,v)\right|\to0. \tag*{(134)} u , v ∈ L M hard sup ( uv ) − d 1 /2 ( κ N ) 2 1 K N ( a , b ) ( ( κ N ) 2 u , ( κ N ) 2 v ; q ) − K d 1 Bessel ( u , v ) → 0. (134) In particular, if d 1 ≥ 0 d_1\ge0 d 1 ≥ 0 ( u v ) − d 1 / 2 (uv)^{-d_1/2} ( uv ) − d 1 /2
Proof. Comparing coefficients in the basic hypergeometric representation, we obtain
p N ( a , b ) ( x ; q ) − p N − 1 ( a , b ) ( x ; q ) = − q 1 − N ( 1 − q 2 N + a + b ) 1 − q a + 1 x p N − 1 ( a + 1 , b ) ( x ; q ) . (135) p_N^{(a,b)}(x;q)-p_{N-1}^{(a,b)}(x;q)
=-\frac{q^{1-N}(1-q^{2N+a+b})}{1-q^{a+1}}xp_{N-1}^{(a+1,b)}(x;q). \tag*{(135)} p N ( a , b ) ( x ; q ) − p N − 1 ( a , b ) ( x ; q ) = − 1 − q a + 1 q 1 − N ( 1 − q 2 N + a + b ) x p N − 1 ( a + 1 , b ) ( x ; q ) . (135) For convenience, write
f N ( u ) = p N ( a , b ) ( u ( κ N ) 2 ; q ) , g N ( u ) = u p N − 1 ( a + 1 , b ) ( u ( κ N ) 2 ; q ) . f_N(u)=p_N^{(a,b)}\left(\frac{u}{(\kappa N)^2};q\right),\qquad
g_N(u)=up_{N-1}^{(a+1,b)}\left(\frac{u}{(\kappa N)^2};q\right). f N ( u ) = p N ( a , b ) ( ( κ N ) 2 u ; q ) , g N ( u ) = u p N − 1 ( a + 1 , b ) ( ( κ N ) 2 u ; q ) . Then we have
( κ N ) 2 u − v [ p N ( a , b ) ( u ( κ N ) 2 ; q ) p N − 1 ( a , b ) ( v ( κ N ) 2 ; q ) − p N − 1 ( a , b ) ( u ( κ N ) 2 ; q ) p N ( a , b ) ( v ( κ N ) 2 ; q ) ] = q 1 − N ( 1 − q 2 N + a + b ) 1 − q a + 1 f N ( u ) g N ( v ) − g N ( u ) f N ( v ) u − v . \frac{(\kappa N)^2}{u-v}\left[
p_N^{(a,b)}\left(\frac{u}{(\kappa N)^2};q\right)
p_{N-1}^{(a,b)}\left(\frac{v}{(\kappa N)^2};q\right)
-p_{N-1}^{(a,b)}\left(\frac{u}{(\kappa N)^2};q\right)
p_N^{(a,b)}\left(\frac{v}{(\kappa N)^2};q\right)
\right]
=
\frac{q^{1-N}(1-q^{2N+a+b})}{1-q^{a+1}}
\frac{f_N(u)g_N(v)-g_N(u)f_N(v)}{u-v}. u − v ( κ N ) 2 [ p N ( a , b ) ( ( κ N ) 2 u ; q ) p N − 1 ( a , b ) ( ( κ N ) 2 v ; q ) − p N − 1 ( a , b ) ( ( κ N ) 2 u ; q ) p N ( a , b ) ( ( κ N ) 2 v ; q ) ] = 1 − q a + 1 q 1 − N ( 1 − q 2 N + a + b ) u − v f N ( u ) g N ( v ) − g N ( u ) f N ( v ) . By (6.7), applied with parameters a a a a + 1 a+1 a + 1
f N ( u ) = Γ ( d 1 + 1 ) 2 d 1 u − d 1 / 2 J d 1 ( u ) + o ( 1 ) , f_N(u)=\Gamma(d_1+1)2^{d_1}u^{-d_1/2}J_{d_1}(\sqrt{u})+o(1), f N ( u ) = Γ ( d 1 + 1 ) 2 d 1 u − d 1 /2 J d 1 ( u ) + o ( 1 ) , g N ( u ) = Γ ( d 1 + 2 ) 2 d 1 + 1 u − d 1 / 2 u J d 1 + 1 ( u ) + o ( 1 ) , g_N(u)=\Gamma(d_1+2)2^{d_1+1}u^{-d_1/2}\sqrt{u}J_{d_1+1}(\sqrt{u})+o(1), g N ( u ) = Γ ( d 1 + 2 ) 2 d 1 + 1 u − d 1 /2 u J d 1 + 1 ( u ) + o ( 1 ) , locally uniformly in u u u u u u v v v
f N ( u ) g N ( v ) − g N ( u ) f N ( v ) u − v = g N ( u ) ∫ 0 1 f N ′ ( v + t ( u − v ) ) d t − f N ( u ) ∫ 0 1 g N ′ ( v + t ( u − v ) ) d t . \frac{f_N(u)g_N(v)-g_N(u)f_N(v)}{u-v}
=g_N(u)\int_0^1 f_N'\bigl(v+t(u-v)\bigr)\,dt-f_N(u)\int_0^1 g_N'\bigl(v+t(u-v)\bigr)\,dt. u − v f N ( u ) g N ( v ) − g N ( u ) f N ( v ) = g N ( u ) ∫ 0 1 f N ′ ( v + t ( u − v ) ) d t − f N ( u ) ∫ 0 1 g N ′ ( v + t ( u − v ) ) d t . The right-hand side is defined also at u = v u=v u = v [ 0 , M ] 2 [0,M]^2 [ 0 , M ] 2
Using
q 1 − N ( 1 − q 2 N + a + b ) 1 − q a + 1 = N λ 1 − s c 2 + 2 s ( d 1 + 1 ) ( 1 + O ( N − 1 ) ) \frac{q^{1-N}\left(1-q^{2N+a+b}\right)}{1-q^{a+1}}
=\frac{N}{\lambda}\frac{1-s^{c_2+2}}{s(d_1+1)}\left(1+O(N^{-1})\right) 1 − q a + 1 q 1 − N ( 1 − q 2 N + a + b ) = λ N s ( d 1 + 1 ) 1 − s c 2 + 2 ( 1 + O ( N − 1 ) ) and the Bessel identity [86 Eq. (10.6.2)]
J ν ′ ( z ) = − J ν + 1 ( z ) + ν z J ν ( z ) , J_\nu'(z)=-J_{\nu+1}(z)+\frac{\nu}{z}J_\nu(z), J ν ′ ( z ) = − J ν + 1 ( z ) + z ν J ν ( z ) , we obtain
q 1 − N ( 1 − q 2 N + a + b ) 1 − q a + 1 f N ( u ) g N ( v ) − g N ( u ) f N ( v ) u − v = − N λ 1 − s c 2 + 2 s Γ ( d 1 + 1 ) 2 2 2 d 1 + 2 ( ( u v ) − d 1 / 2 K d 1 B e s s e l ( u , v ) + o ( 1 ) ) . (136) \frac{q^{1-N}\left(1-q^{2N+a+b}\right)}{1-q^{a+1}}
\frac{f_N(u)g_N(v)-g_N(u)f_N(v)}{u-v}
=-\frac{N}{\lambda}\frac{1-s^{c_2+2}}{s}\Gamma(d_1+1)^2 2^{2d_1+2}
\left((uv)^{-d_1/2}K_{d_1}^{\mathrm{Bessel}}(u,v)+o(1)\right). \tag*{(136)} 1 − q a + 1 q 1 − N ( 1 − q 2 N + a + b ) u − v f N ( u ) g N ( v ) − g N ( u ) f N ( v ) = − λ N s 1 − s c 2 + 2 Γ ( d 1 + 1 ) 2 2 2 d 1 + 2 ( ( uv ) − d 1 /2 K d 1 Bessel ( u , v ) + o ( 1 ) ) . (136) The error is uniform on [ 0 , M ] 2 [0,M]^2 [ 0 , M ] 2
The explicit formulae for the squared norms and leading coefficients give
h N − 1 ( a , b ) = s d 1 + 1 Γ ( d 1 + 1 ) 2 ( 1 − s ) d 1 ( 1 − s c 2 + 1 ) d 1 ( 1 − s c 2 + 2 ) ( λ N ) 2 d 1 + 1 ( 1 + O ( N − 1 ) ) , h_{N-1}^{(a,b)}=
\frac{s^{d_1+1}\Gamma(d_1+1)^2}
{(1-s)^{d_1}(1-sc_2+1)^{d_1}(1-sc_2+2)}
\left(\frac{\lambda}{N}\right)^{2d_1+1}
\left(1+O(N^{-1})\right), h N − 1 ( a , b ) = ( 1 − s ) d 1 ( 1 − s c 2 + 1 ) d 1 ( 1 − s c 2 + 2 ) s d 1 + 1 Γ ( d 1 + 1 ) 2 ( N λ ) 2 d 1 + 1 ( 1 + O ( N − 1 ) ) , γ N − 1 ( a , b ) γ N ( a , b ) = − s ( 1 − s ) ( 1 − s c 2 + 1 ) ( 1 − s c 2 + 2 ) 2 + O ( N − 1 ) . (137) \frac{\gamma_{N-1}^{(a,b)}}{\gamma_N^{(a,b)}}
=-\frac{s(1-s)(1-sc_2+1)}
{(1-sc_2+2)^2}+O(N^{-1}). \tag*{(137)} γ N ( a , b ) γ N − 1 ( a , b ) = − ( 1 − s c 2 + 2 ) 2 s ( 1 − s ) ( 1 − s c 2 + 1 ) + O ( N − 1 ) . (137) On the other hand, it follows from (3.4) that
( w ( a , b ) ( u ( κ N ) 2 ; q ) w ( a , b ) ( v ( κ N ) 2 ; q ) ) 1 / 2 = ( u v ( κ N ) 4 ) d 1 / 2 ( 1 + O ( N − 1 ) ) (138) \left(w^{(a,b)}\left(\frac{u}{(\kappa N)^2};q\right)
w^{(a,b)}\left(\frac{v}{(\kappa N)^2};q\right)\right)^{1/2}
=\left(\frac{uv}{(\kappa N)^4}\right)^{d_1/2}
\left(1+O(N^{-1})\right) \tag*{(138)} ( w ( a , b ) ( ( κ N ) 2 u ; q ) w ( a , b ) ( ( κ N ) 2 v ; q ) ) 1/2 = ( ( κ N ) 4 uv ) d 1 /2 ( 1 + O ( N − 1 ) ) (138) uniformly for 0 < u , v ≤ M 0<u,v\le M 0 < u , v ≤ M
Substituting (6.11), (6.12) and (6.13) into the Christoffel–Darboux formula (6.5), and using (3.31), the proposition follows. □
We now pass from the weighted uniform convergence of the correlation kernels to convergence of the associated Fredholm determinants. As in the proof of Proposition 4.1, it is convenient to regard the q q q Z ≥ 0 \mathbb{Z}_{\ge0} Z ≥ 0
K N ( a , b ) ( k , l ; q ) = ( 1 − q ) q ( k + l ) / 2 K N ( a , b ) ( q k , q l ; q ) . (139) K_N^{(a,b)}(k,l;q)=(1-q)q^{(k+l)/2}K_N^{(a,b)}(q^k,q^l;q). \tag*{(139)} K N ( a , b ) ( k , l ; q ) = ( 1 − q ) q ( k + l ) /2 K N ( a , b ) ( q k , q l ; q ) . (139) For a fixed s > 0 s>0 s > 0
I N ( s ) = { k ∈ Z ≥ 0 : ( κ N ) 2 q k ≤ s } . (140) I_N(s)=\{k\in\mathbb{Z}_{\ge0}:(\kappa N)^2q^k\le s\}. \tag*{(140)} I N ( s ) = { k ∈ Z ≥ 0 : ( κ N ) 2 q k ≤ s } . (140) Then we have the following.
Proposition 6.3. Let q = e − λ / N q=e^{-\lambda/N} q = e − λ / N for a fixed λ > 0 \lambda>0 λ > 0 and let a = d 1 > − 1 a=d_1>-1 a = d 1 > − 1 and b = c 2 N + d 2 b=c_2N+d_2 b = c 2 N + d 2 Then, for each fixed s > 0 s>0 s > 0
lim N → ∞ det ( Id − K N ( a , b ) ) ℓ 2 ( I N ( s ) ) = det ( Id − K d 1 B e s s e l ) L 2 ( 0 , s ) . (141) \lim_{N\to\infty}\det\left(\operatorname{Id}-K_N^{(a,b)}\right)_{\ell^2(I_N(s))}
=\det\left(\operatorname{Id}-K_{d_1}^{\mathrm{Bessel}}\right)_{L^2(0,s)}. \tag*{(141)} N → ∞ lim det ( Id − K N ( a , b ) ) ℓ 2 ( I N ( s )) = det ( Id − K d 1 Bessel ) L 2 ( 0 , s ) . (141) Proof. Write
u k = ( κ N ) 2 q k , H ( u , v ) = ( u v ) − d 1 / 2 K d 1 B e s s e l ( u , v ) . u_k=(\kappa N)^2q^k,\qquad H(u,v)=(uv)^{-d_1/2}K_{d_1}^{\mathrm{Bessel}}(u,v). u k = ( κ N ) 2 q k , H ( u , v ) = ( uv ) − d 1 /2 K d 1 Bessel ( u , v ) . The power series of the Bessel function shows that H H H [ 0 , s ] 2 [0,s]^2 [ 0 , s ] 2
K N ( a , b ) ( k , l ; q ) = ( 1 − q ) ( u k u l ) ( d 1 + 1 ) / 2 ( H ( u k , u l ) + o ( 1 ) ) , (142) K_N^{(a,b)}(k,l;q)=(1-q)(u_ku_l)^{(d_1+1)/2}\left(H(u_k,u_l)+o(1)\right), \tag*{(142)} K N ( a , b ) ( k , l ; q ) = ( 1 − q ) ( u k u l ) ( d 1 + 1 ) /2 ( H ( u k , u l ) + o ( 1 ) ) , (142) uniformly for k , l ∈ I N ( s ) k,l\in I_N(s) k , l ∈ I N ( s ) K N ( a , b ) K_N^{(a,b)} K N ( a , b ) N N N ℓ 2 ( Z ≥ 0 ) \ell^2(\mathbb{Z}_{\geq0}) ℓ 2 ( Z ≥ 0 ) ℓ 2 ( I N ( s ) ) \ell^2(I_N(s)) ℓ 2 ( I N ( s )) H H H [ 0 , s ] 2 [0,s]^2 [ 0 , s ] 2 C > 0 C>0 C > 0
0 ≤ K N ( a , b ) ( k , k ; q ) ≤ C ( 1 − q ) u k d 1 + 1 , k ∈ I N ( s ) , 0\leq K_N^{(a,b)}(k,k;q)\leq C(1-q)u_k^{d_1+1},\qquad k\in I_N(s), 0 ≤ K N ( a , b ) ( k , k ; q ) ≤ C ( 1 − q ) u k d 1 + 1 , k ∈ I N ( s ) , for all sufficiently large N N N
∑ k ∈ I N ( s ) K N ( a , b ) ( k , k ; q ) ≤ C ( 1 − q ) ∑ k ∈ I N ( s ) u k d 1 + 1 = C 1 − q 1 − q d 1 + 1 u min I N ( s ) d 1 + 1 ≤ C s , (143) \sum_{k\in I_N(s)}K_N^{(a,b)}(k,k;q)\leq C(1-q)\sum_{k\in I_N(s)}u_k^{d_1+1}
=C\frac{1-q}{1-q^{d_1+1}}u_{\min I_N(s)}^{d_1+1}\leq C_s, \tag*{(143)} k ∈ I N ( s ) ∑ K N ( a , b ) ( k , k ; q ) ≤ C ( 1 − q ) k ∈ I N ( s ) ∑ u k d 1 + 1 = C 1 − q d 1 + 1 1 − q u m i n I N ( s ) d 1 + 1 ≤ C s , (143) where C s C_s C s N N N u min I N ( s ) ≤ s u_{\min I_N(s)}\leq s u m i n I N ( s ) ≤ s
We next prove convergence of each term in the Fredholm expansion. For a fixed m ≥ 1 m\geq1 m ≥ 1 u k j u_{k_j} u k j
det [ K N ( a , b ) ( k j , k l ; q ) ] j , l = 1 m = ∏ j = 1 m ( ( 1 − q ) u k j d 1 + 1 ) ( det [ H ( u k j , u k l ) ] j , l = 1 m + o ( 1 ) ) , \det\left[K_N^{(a,b)}(k_j,k_l;q)\right]_{j,l=1}^{m}
=\prod_{j=1}^{m}\left((1-q)u_{k_j}^{d_1+1}\right)\left(\det\left[H(u_{k_j},u_{k_l})\right]_{j,l=1}^{m}+o(1)\right), det [ K N ( a , b ) ( k j , k l ; q ) ] j , l = 1 m = j = 1 ∏ m ( ( 1 − q ) u k j d 1 + 1 ) ( det [ H ( u k j , u k l ) ] j , l = 1 m + o ( 1 ) ) , uniformly for k 1 , … , k m ∈ I N ( s ) k_1,\ldots,k_m\in I_N(s) k 1 , … , k m ∈ I N ( s )
∑ k 1 , … , k m ∈ I N ( s ) ∏ j = 1 m ( ( 1 − q ) u k j d 1 + 1 ) = ( 1 − q 1 − q d 1 + 1 u min I N ( s ) d 1 + 1 ) m \sum_{k_1,\ldots,k_m\in I_N(s)}\prod_{j=1}^{m}\left((1-q)u_{k_j}^{d_1+1}\right)
=\left(\frac{1-q}{1-q^{d_1+1}}u_{\min I_N(s)}^{d_1+1}\right)^m k 1 , … , k m ∈ I N ( s ) ∑ j = 1 ∏ m ( ( 1 − q ) u k j d 1 + 1 ) = ( 1 − q d 1 + 1 1 − q u m i n I N ( s ) d 1 + 1 ) m is bounded independently of N N N o ( 1 ) o(1) o ( 1 )
To identify the limit of the leading term, associate to each k ∈ I N ( s ) k\in I_N(s) k ∈ I N ( s ) ( u k + 1 , u k ] (u_{k+1},u_k] ( u k + 1 , u k ]
⋃ k ∈ I N ( s ) ( u k + 1 , u k ] = ( 0 , u min I N ( s ) ] , q s < u min I N ( s ) ≤ s \bigcup_{k\in I_N(s)}(u_{k+1},u_k]=(0,u_{\min I_N(s)}],\qquad qs<u_{\min I_N(s)}\leq s k ∈ I N ( s ) ⋃ ( u k + 1 , u k ] = ( 0 , u m i n I N ( s ) ] , q s < u m i n I N ( s ) ≤ s for all sufficiently large $N, and their maximal length is at most ( 1 − q ) s (1-q)s ( 1 − q ) s
( 1 − q ) u k d 1 + 1 = ( d 1 + 1 ) ( 1 − q ) 1 − q d 1 + 1 ∫ u k + 1 u k u d 1 d u , (1-q)u_k^{d_1+1}=\frac{(d_1+1)(1-q)}{1-q^{d_1+1}}\int_{u_{k+1}}^{u_k}u^{d_1}\,du, ( 1 − q ) u k d 1 + 1 = 1 − q d 1 + 1 ( d 1 + 1 ) ( 1 − q ) ∫ u k + 1 u k u d 1 d u , where the prefactor tends to one. Thus these sums are Riemann sums with respect to the finite measure u d 1 d u u^{d_1}\,du u d 1 d u ( 0 , s ) (0,s) ( 0 , s ) H H H [ 0 , s ] 2 [0,s]^2 [ 0 , s ] 2
lim N → ∞ ∑ k 1 , … , k m ∈ I N ( s ) det [ K N ( a , b ) ( k j , k l ; q ) ] j , l = 1 m = ∫ ( 0 , s ) m det [ H ( u j , u l ) ] j , l = 1 m ∏ j = 1 m u j d 1 d u j = ∫ ( 0 , s ) m det [ K d 1 B e s s e l ( u j , u l ) ] j , l = 1 m d u 1 ⋯ d u m . (144) \begin{aligned}
\lim_{N\to\infty}\sum_{k_1,\ldots,k_m\in I_N(s)}
\det\left[K_N^{(a,b)}(k_j,k_l;q)\right]_{j,l=1}^{m}
&=\int_{(0,s)^m}\det\left[H(u_j,u_l)\right]_{j,l=1}^{m}\prod_{j=1}^{m}u_j^{d_1}\,du_j\\
&=\int_{(0,s)^m}\det\left[K_{d_1}^{\mathrm{Bessel}}(u_j,u_l)\right]_{j,l=1}^{m}\,du_1\cdots du_m. \tag*{(144)}
\end{aligned} N → ∞ lim k 1 , … , k m ∈ I N ( s ) ∑ det [ K N ( a , b ) ( k j , k l ; q ) ] j , l = 1 m = ∫ ( 0 , s ) m det [ H ( u j , u l ) ] j , l = 1 m j = 1 ∏ m u j d 1 d u j = ∫ ( 0 , s ) m det [ K d 1 Bessel ( u j , u l ) ] j , l = 1 m d u 1 ⋯ d u m . (144) It remains to interchange the limit with the Fredholm series. We have
det ( Id − K N ( a , b ) ) ℓ 2 ( I N ( s ) ) = ∑ m = 0 ∞ ( − 1 ) m m ! ∑ k 1 , … , k m ∈ I N ( s ) det [ K N ( a , b ) ( k j , k l ; q ) ] j , l = 1 m , (145) \det\left(\operatorname{Id}-K_N^{(a,b)}\right)_{\ell^2(I_N(s))}
=\sum_{m=0}^{\infty}\frac{(-1)^m}{m!}
\sum_{k_1,\ldots,k_m\in I_N(s)}
\det\left[K_N^{(a,b)}(k_j,k_l;q)\right]_{j,l=1}^{m}, \tag*{(145)} det ( Id − K N ( a , b ) ) ℓ 2 ( I N ( s )) = m = 0 ∑ ∞ m ! ( − 1 ) m k 1 , … , k m ∈ I N ( s ) ∑ det [ K N ( a , b ) ( k j , k l ; q ) ] j , l = 1 m , (145) with the m = 0 m=0 m = 0
0 ≤ det [ K N ( a , b ) ( k j , k l ; q ) ] j , l = 1 m ≤ ∏ j = 1 m K N ( a , b ) ( k j , k j ; q ) . 0\leq\det\left[K_N^{(a,b)}(k_j,k_l;q)\right]_{j,l=1}^{m}
\leq\prod_{j=1}^{m}K_N^{(a,b)}(k_j,k_j;q). 0 ≤ det [ K N ( a , b ) ( k j , k l ; q ) ] j , l = 1 m ≤ j = 1 ∏ m K N ( a , b ) ( k j , k j ; q ) . Hence, by (6.18),
1 m ! ∑ k 1 , … , k m ∈ I N ( s ) det [ K N ( a , b ) ( k j , k l ; q ) ] j , l = 1 m ≤ 1 m ! ( ∑ k ∈ I N ( s ) K N ( a , b ) ( k , k ; q ) ) m ≤ C s m m ! . \frac{1}{m!}\sum_{k_1,\ldots,k_m\in I_N(s)}\det\left[K_N^{(a,b)}(k_j,k_l;q)\right]_{j,l=1}^{m}\leq\frac{1}{m!}\left(\sum_{k\in I_N(s)}K_N^{(a,b)}(k,k;q)\right)^m\leq\frac{C_s^m}{m!}. m ! 1 k 1 , … , k m ∈ I N ( s ) ∑ det [ K N ( a , b ) ( k j , k l ; q ) ] j , l = 1 m ≤ m ! 1 k ∈ I N ( s ) ∑ K N ( a , b ) ( k , k ; q ) m ≤ m ! C s m . This bound is summable over m m m N N N
Proof of Theorem 3.2 (i). Fix s > 0 s>0 s > 0 I N ( s ) I_N(s) I N ( s ) x min > s / ( κ N ) 2 x_{\min}>s/(\kappa N)^2 x m i n > s / ( κ N ) 2 q k q^k q k k ∈ I N ( s ) k\in I_N(s) k ∈ I N ( s )
P ( x min > s ( κ N ) 2 ) = det ( Id − K N ( a , b ) ) ℓ 2 ( I N ( s ) ) . \mathbb{P}\left(x_{\min}>\frac{s}{(\kappa N)^2}\right)=\det\left(\operatorname{Id}-K_N^{(a,b)}\right)_{\ell^2(I_N(s))}. P ( x m i n > ( κ N ) 2 s ) = det ( Id − K N ( a , b ) ) ℓ 2 ( I N ( s )) . Taking N → ∞ N\to\infty N → ∞
Soft edge scaling limit
We now turn to the soft-edge regime c 1 > 0 c_1>0 c 1 > 0 a a a θ \theta θ u , v ∈ R u,v\in\mathbb{R} u , v ∈ R
x ^ = a − u ( θ N ) 2 / 3 , y ^ = a − v ( θ N ) 2 / 3 . (146) \widehat{x}=a-\frac{u}{(\theta N)^{2/3}},\qquad\widehat{y}=a-\frac{v}{(\theta N)^{2/3}}. \tag*{(146)} x = a − ( θ N ) 2/3 u , y = a − ( θ N ) 2/3 v . (146) Proposition 6.4. Let q = e − λ / N q=e^{-\lambda/N} q = e − λ / N λ > 0 \lambda>0 λ > 0 c 1 > 0 c_1>0 c 1 > 0 c 2 ≥ 0 c_2\geq0 c 2 ≥ 0 d 2 > − 1 d_2>-1 d 2 > − 1 M > 0 M>0 M > 0
L M s o f t = { u ∈ R : ∣ u ∣ < M , a − u ( θ N ) 2 / 3 ∈ L q } , (147) L_M^{\mathrm{soft}}=\left\{u\in\mathbb{R}: |u|<M,\ a-\frac{u}{(\theta N)^{2/3}}\in L_q\right\}, \tag*{(147)} L M soft = { u ∈ R : ∣ u ∣ < M , a − ( θ N ) 2/3 u ∈ L q } , (147) where L q = { 1 , q , q 2 , … } L_q=\{1,q,q^2,\ldots\} L q = { 1 , q , q 2 , … } N → ∞ , N\to\infty, N → ∞ ,
sup u , v ∈ L M s o f t ∣ 1 ( θ N ) 2 / 3 K N ( a , b ) ( x ^ , y ^ ; q ) − K A i ( u , v ) ∣ → 0. (148) \sup_{u,v\in L_M^{\mathrm{soft}}}\left|\frac{1}{(\theta N)^{2/3}}K_N^{(a,b)}(\widehat{x},\widehat{y};q)-K_{\mathrm{Ai}}(u,v)\right|\to0. \tag*{(148)} u , v ∈ L M soft sup ( θ N ) 2/3 1 K N ( a , b ) ( x , y ; q ) − K Ai ( u , v ) → 0. (148) To prove Proposition 6.4, we apply the method of steepest descent to a contour integral representation of the little q q q n ∈ Z ≥ 0 n\in\mathbb{Z}_{\geq0} n ∈ Z ≥ 0
F ~ n ( x , z ) : = z n ( − q / z , − q n + a + b + 2 x / z ; q ) ∞ ( − q x / z , − q n + a + 1 / z ; q ) ∞ . (149) \widetilde{F}_n(x,z):=z^n\frac{(-q/z,-q^{n+a+b+2}x/z;q)_\infty}{(-qx/z,-q^{n+a+1}/z;q)_\infty}. \tag*{(149)} F n ( x , z ) := z n ( − q x / z , − q n + a + 1 / z ; q ) ∞ ( − q / z , − q n + a + b + 2 x / z ; q ) ∞ . (149) For fixed x ∈ ( 0 , 1 ) x\in(0,1) x ∈ ( 0 , 1 ) F ~ n ( x , z ) \widetilde{F}_n(x,z) F n ( x , z ) z z z
− q n + a + 1 + j , − x q 1 + j , j ∈ Z ≥ 0 . -q^{n+a+1+j},\qquad-xq^{1+j},\qquad j\in\mathbb{Z}_{\geq0}. − q n + a + 1 + j , − x q 1 + j , j ∈ Z ≥ 0 . The following representation provides the starting point for our asymptotic analysis.
Lemma 6.5. For n ∈ Z ≥ 0 n\in\mathbb{Z}_{\geq0} n ∈ Z ≥ 0 x ∈ ( 0 , 1 ) x\in(0,1) x ∈ ( 0 , 1 )
p n ( a , b ) ( x ; q ) = ( q ; q ) n ( q a + 1 ; q ) n q − n ( n + 1 ) / 2 ∮ ∣ z ∣ = r F ~ n ( x , z ) d z 2 π i z , (150) p_n^{(a,b)}(x;q)=\frac{(q;q)_n}{(q^{a+1};q)_n}q^{-n(n+1)/2}\oint_{|z|=r}\widetilde{F}_n(x,z)\frac{dz}{2\pi iz}, \tag*{(150)} p n ( a , b ) ( x ; q ) = ( q a + 1 ; q ) n ( q ; q ) n q − n ( n + 1 ) /2 ∮ ∣ z ∣ = r F n ( x , z ) 2 π i z d z , (150) where the circle is positively oriented and its radius satisfies max { q n + a + 1 , q x } < r < 1 \max\{q^{n+a+1},qx\}<r<1 max { q n + a + 1 , q x } < r < 1
Proof. We use the coefficient integral [63 ] , Eq. (3.3),
( A ; q ) k ( B ; q ) k = ( q , B / A ; q ) ∞ ( B , q / A ; q ) ∞ ∮ ∣ z ∣ = r ( A z , q / ( A z ) ; q ) ∞ ( z , B / ( A z ) ; q ) ∞ d z 2 π i z k + 1 , ∣ B / A ∣ < r < 1. (151) \frac{(A;q)_k}{(B;q)_k}=\frac{(q,B/A;q)_\infty}{(B,q/A;q)_\infty}\oint_{|z|=r}\frac{(Az,q/(Az);q)_\infty}{(z,B/(Az);q)_\infty}\frac{dz}{2\pi iz^{k+1}},\qquad|B/A|<r<1. \tag*{(151)} ( B ; q ) k ( A ; q ) k = ( B , q / A ; q ) ∞ ( q , B / A ; q ) ∞ ∮ ∣ z ∣ = r ( z , B / ( A z ) ; q ) ∞ ( A z , q / ( A z ) ; q ) ∞ 2 π i z k + 1 d z , ∣ B / A ∣ < r < 1. (151) We apply this identity with A = q − n A=q^{-n} A = q − n B = q a + 1 B=q^{a+1} B = q a + 1 max { q n + a + 1 , q x } < r < 1 \max\{q^{n+a+1},qx\}<r<1 max { q n + a + 1 , q x } < r < 1 ( q − n ; q ) k = 0 (q^{-n};q)_k=0 ( q − n ; q ) k = 0 k > n k>n k > n k ≥ 0 k\geq0 k ≥ 0 ∣ z ∣ = r |z|=r ∣ z ∣ = r ∣ q x / z ∣ < 1 |qx/z|<1 ∣ q x / z ∣ < 1 q q q
p n ( a , b ) ( x ; q ) = ( q ; q ) n ( q a + 1 ; q ) n ∮ ∣ z ∣ = r ( q − n z ; q ) n ( q n + 1 / z ; q ) ∞ ( q n + a + 1 / z ; q ) ∞ ∑ k = 0 ∞ ( q n + a + b + 1 ; q ) k ( q ; q ) k ( q x z ) k d z 2 π i z = ( q ; q ) n ( q a + 1 ; q ) n ∮ ∣ z ∣ = r ( q − n z ; q ) n ( q n + 1 / z , q n + a + b + 2 x / z ; q ) ∞ ( q n + a + 1 / z , q x / z ; q ) ∞ d z 2 π i z . \begin{aligned}
p_n^{(a,b)}(x;q)=\frac{(q;q)_n}{(q^{a+1};q)_n}\oint_{|z|=r}\frac{(q^{-n}z;q)_n(q^{n+1}/z;q)_\infty}{(q^{n+a+1}/z;q)_\infty}\sum_{k=0}^{\infty}\frac{(q^{n+a+b+1};q)_k}{(q;q)_k}\left(\frac{qx}{z}\right)^k\frac{dz}{2\pi iz} \\
&=\frac{(q;q)_n}{(q^{a+1};q)_n}\oint_{|z|=r}\frac{(q^{-n}z;q)_n(q^{n+1}/z,q^{n+a+b+2}x/z;q)_\infty}{(q^{n+a+1}/z,qx/z;q)_\infty}\frac{dz}{2\pi iz}.
\end{aligned} p n ( a , b ) ( x ; q ) = ( q a + 1 ; q ) n ( q ; q ) n ∮ ∣ z ∣ = r ( q n + a + 1 / z ; q ) ∞ ( q − n z ; q ) n ( q n + 1 / z ; q ) ∞ k = 0 ∑ ∞ ( q ; q ) k ( q n + a + b + 1 ; q ) k ( z q x ) k 2 π i z d z = ( q a + 1 ; q ) n ( q ; q ) n ∮ ∣ z ∣ = r ( q n + a + 1 / z , q x / z ; q ) ∞ ( q − n z ; q ) n ( q n + 1 / z , q n + a + b + 2 x / z ; q ) ∞ 2 π i z d z . where the second equality follows from the q q q
( q − n z ; q ) n = ( − z ) n q − n ( n + 1 ) / 2 ( q / z ; q ) n , ( q / z ; q ) n ( q n + 1 / z ; q ) ∞ = ( q / z ; q ) ∞ , (q^{-n}z;q)_n=(-z)^nq^{-n(n+1)/2}(q/z;q)_n,\qquad(q/z;q)_n(q^{n+1}/z;q)_\infty=(q/z;q)_\infty, ( q − n z ; q ) n = ( − z ) n q − n ( n + 1 ) /2 ( q / z ; q ) n , ( q / z ; q ) n ( q n + 1 / z ; q ) ∞ = ( q / z ; q ) ∞ , and making the change of variables z ↦ − z z\mapsto-z z ↦ − z (150) . □
In view of (150) , we introduce the normalised polynomial
p n ( x ) : = ∮ ∣ z ∣ = r F ~ n ( x , z ) d z 2 π i z = ( q a + 1 ; q ) n ( q ; q ) n q n ( n + 1 ) / 2 p n ( a , b ) ( x ; q ) . (152) p_n(x):=\oint_{|z|=r}\widetilde{F}_n(x,z)\frac{dz}{2\pi iz}=\frac{(q^{a+1};q)_n}{(q;q)_n}q^{n(n+1)/2}p_n^{(a,b)}(x;q). \tag*{(152)} p n ( x ) := ∮ ∣ z ∣ = r F n ( x , z ) 2 π i z d z = ( q ; q ) n ( q a + 1 ; q ) n q n ( n + 1 ) /2 p n ( a , b ) ( x ; q ) . (152) where r r r (150) . The Christoffel–Darboux formula (129) then becomes
K N ( a , b ) ( x , y ; q ) = C N ( a , b ) w ( a , b ) ( x ; q ) w ( a , b ) ( y ; q ) p N − 1 ( x ) p N ( y ) − p N ( x ) p N − 1 ( y ) x − y , (153) K_N^{(a,b)}(x,y;q)=C_N^{(a,b)}\sqrt{w^{(a,b)}(x;q)w^{(a,b)}(y;q)}\frac{p_{N-1}(x)p_N(y)-p_N(x)p_{N-1}(y)}{x-y}, \tag*{(153)} K N ( a , b ) ( x , y ; q ) = C N ( a , b ) w ( a , b ) ( x ; q ) w ( a , b ) ( y ; q ) x − y p N − 1 ( x ) p N ( y ) − p N ( x ) p N − 1 ( y ) , (153) with the positive normalisation constant
C N ( a , b ) = q − N 2 − a ( N − 1 ) ( 1 − q ) ( 1 − q 2 N + a + b ) ( q a + N , q b + N ; q ) ∞ ( q N + 1 , q a + b + N + 1 ; q ) ∞ . C_N^{(a,b)}=\frac{q^{-N^2-a(N-1)}}{(1-q)(1-q^{2N+a+b})}\frac{(q^{a+N},q^{b+N};q)_\infty}{(q^{N+1},q^{a+b+N+1};q)_\infty}. C N ( a , b ) = ( 1 − q ) ( 1 − q 2 N + a + b ) q − N 2 − a ( N − 1 ) ( q N + 1 , q a + b + N + 1 ; q ) ∞ ( q a + N , q b + N ; q ) ∞ . To analyse the contour integral, we introduce the phase and amplitude functions
Φ ( x , z ) : = log z + 1 λ ( Li 2 ( − x / z ) + Li 2 ( − s c 1 + 1 / z ) − Li 2 ( − 1 / z ) − Li 2 ( − s c 1 + c 2 + 1 x / z ) ) , (154) \Phi(x,z):=\log z+\frac{1}{\lambda}\left(\operatorname{Li}_2(-x/z)+\operatorname{Li}_2(-s^{c_1+1}/z)-\operatorname{Li}_2(-1/z)-\operatorname{Li}_2(-s^{c_1+c_2+1}x/z)\right), \tag*{(154)} Φ ( x , z ) := log z + λ 1 ( Li 2 ( − x / z ) + Li 2 ( − s c 1 + 1 / z ) − Li 2 ( − 1/ z ) − Li 2 ( − s c 1 + c 2 + 1 x / z ) ) , (154) Ψ ( x , z ) : = ( x + z 1 + z ) 1 / 2 ( 1 + s c 1 + 1 z ) d 1 + 1 / 2 ( 1 + s c 1 + c 2 + 1 x z ) − ( d 1 + d 2 + 3 / 2 ) . (155) \Psi(x,z):=\left(\frac{x+z}{1+z}\right)^{1/2}\left(1+\frac{s^{c_1+1}}{z}\right)^{d_1+1/2}\left(1+\frac{s^{c_1+c_2+1}x}{z}\right)^{-(d_1+d_2+3/2)}. \tag*{(155)} Ψ ( x , z ) := ( 1 + z x + z ) 1/2 ( 1 + z s c 1 + 1 ) d 1 + 1/2 ( 1 + z s c 1 + c 2 + 1 x ) − ( d 1 + d 2 + 3/2 ) . (155) The branches are chosen so that Φ \Phi Φ Ψ \Psi Ψ x , z > 0 x,z>0 x , z > 0 F ~ N \widetilde{F}_N F N Φ z = ∂ z Φ \Phi_z=\partial_z\Phi Φ z = ∂ z Φ Φ z z = ∂ z 2 Φ \Phi_{zz}=\partial_z^2\Phi Φ z z = ∂ z 2 Φ Φ x z = ∂ x ∂ z Φ \Phi_{xz}=\partial_x\partial_z\Phi Φ x z = ∂ x ∂ z Φ
It is also convenient to introduce the rational function
ϕ ( x , z ) : = ( x + z ) ( z + s c 1 + 1 ) ( 1 + z ) ( z + s c 1 + c 2 + 1 x ) . \phi(x,z):=\frac{(x+z)(z+s^{c_1+1})}{(1+z)(z+s^{c_1+c_2+1}x)}. ϕ ( x , z ) := ( 1 + z ) ( z + s c 1 + c 2 + 1 x ) ( x + z ) ( z + s c 1 + 1 ) . Direct differentiation gives
s exp [ λ z Φ z ( x , z ) ] = ϕ ( x , z ) . (156) s\exp[\lambda z\Phi_z(x,z)]=\phi(x,z). \tag*{(156)} s exp [ λ z Φ z ( x , z )] = ϕ ( x , z ) . (156) In particular, this identity allows us to locate the relevant saddle point explicitly. Set
z ∗ : = s ( 1 − s c 1 ) − a ( 1 − s c 1 + c 2 + 2 ) 2 ( 1 − s ) . (157) z_*:=\frac{s(1-s^{c_1})-a(1-s^{c_1+c_2+2})}{2(1-s)}. \tag*{(157)} z ∗ := 2 ( 1 − s ) s ( 1 − s c 1 ) − a ( 1 − s c 1 + c 2 + 2 ) . (157) The following lemma identifies the double critical point at the left edge and relates the local derivatives of the phase to the scaling constant θ \theta θ
Lemma 6.6. We have z ∗ > 0 z_*>0 z ∗ > 0 and
Φ z ( a , z ∗ ) = Φ z z ( a , z ∗ ) = 0. (158) \Phi_z(a,z_*)=\Phi_{zz}(a,z_*)=0. \tag*{(158)} Φ z ( a , z ∗ ) = Φ z z ( a , z ∗ ) = 0. (158) Moreover,
z ∗ 2 = s c 1 + 1 a ( 1 − s c 2 + 1 ) 1 − s . (159) z_*^2=\frac{s^{c_1+1}a(1-s^{c_2+1})}{1-s}. \tag*{(159)} z ∗ 2 = 1 − s s c 1 + 1 a ( 1 − s c 2 + 1 ) . (159) and the relevant derivatives satisfy
Φ x z ( a , z ∗ ) = 1 − s c 1 + c 2 + 1 λ ( z ∗ + a ) ( z ∗ + s c 1 + c 2 + 1 a ) > 0 , (160) \Phi_{xz}(a,z_*)=\frac{1-s^{c_1+c_2+1}}{\lambda(z_*+a)(z_*+s^{c_1+c_2+1}a)}>0, \tag*{(160)} Φ x z ( a , z ∗ ) = λ ( z ∗ + a ) ( z ∗ + s c 1 + c 2 + 1 a ) 1 − s c 1 + c 2 + 1 > 0 , (160) Φ z z z ( a , z ∗ ) = 2 ( 1 − s ) λ s z ∗ ( z ∗ + 1 ) ( z ∗ + s c 1 + c 2 + 1 a ) > 0. (161) \Phi_{zzz}(a,z_*)=\frac{2(1-s)}{\lambda s z_*(z_*+1)(z_*+s^{c_1+c_2+1}a)}>0. \tag*{(161)} Φ z z z ( a , z ∗ ) = λ s z ∗ ( z ∗ + 1 ) ( z ∗ + s c 1 + c 2 + 1 a ) 2 ( 1 − s ) > 0. (161) Finally, the soft-edge scaling constant θ \theta θ (64) is related to these derivatives by
θ 2 = 2 Φ x z ( a , z ∗ ) 3 Φ z z z ( a , z ∗ ) . (162) \theta^2=\frac{2\Phi_{xz}(a,z_*)^3}{\Phi_{zzz}(a,z_*)}. \tag*{(162)} θ 2 = Φ z z z ( a , z ∗ ) 2 Φ x z ( a , z ∗ ) 3 . (162) Proof. For x , z > 0 x,z>0 x , z > 0 (156) shows that the saddle equation Φ z ( x , z ) = 0 \Phi_z(x,z)=0 Φ z ( x , z ) = 0 ϕ ( x , z ) = s \phi(x,z)=s ϕ ( x , z ) = s z z z
( 1 − s c 1 + c 2 + 2 ) 2 ( x − a ) ( x − b ) . \left(1-s^{c_1+c_2+2}\right)^2(x-a)(x-b). ( 1 − s c 1 + c 2 + 2 ) 2 ( x − a ) ( x − b ) . By (44) , at x = a x=a x = a z = z ∗ z=z_* z = z ∗
ϕ ( a , z ) − s = ( 1 − s ) ( z − z ∗ ) 2 ( z + 1 ) ( z + s c 1 + c 2 + 1 a ) . (163) \phi(a,z)-s=\frac{(1-s)(z-z_*)^2}{(z+1)(z+s^{c_1+c_2+1}a)}. \tag*{(163)} ϕ ( a , z ) − s = ( z + 1 ) ( z + s c 1 + c 2 + 1 a ) ( 1 − s ) ( z − z ∗ ) 2 . (163) Comparing constant terms gives (159) . The derivative identities follow by differentiating (156) and using (163) . Finally, (162) follows by direct substitution using (44) and (64) . □
We next derive the local asymptotic expansion of the integrand.
Lemma 6.7. Let q = e − λ / N q=e^{-\lambda/N} q = e − λ / N λ > 0 \lambda>0 λ > 0 ( a , z ∗ ) (a,z_*) ( a , z ∗ ) C 2 \mathbb{C}^2 C 2
F ~ N ( x , z ) = Ψ ( x , z ) e N Φ ( x , z ) ( 1 + O ( N − 1 ) ) . (164) \widetilde{F}_N(x,z)=\Psi(x,z)e^{N\Phi(x,z)}(1+O(N^{-1})). \tag*{(164)} F N ( x , z ) = Ψ ( x , z ) e N Φ ( x , z ) ( 1 + O ( N − 1 )) . (164) The relative error and all its partial derivatives of any fixed order are uniformly O ( N − 1 ) O(N^{-1}) O ( N − 1 )
Proof. For fixed d ∈ R d\in\mathbb{R} d ∈ R
log ( z q d ; q ) ∞ = − N λ Li 2 ( z ) + ( 1 2 − d ) log ( 1 − z ) + O ( N − 1 ) , (165) \log(zq^d;q)_\infty=-\frac{N}{\lambda}\operatorname{Li}_2(z)+\left(\frac{1}{2}-d\right)\log(1-z)+O(N^{-1}), \tag*{(165)} log ( z q d ; q ) ∞ = − λ N Li 2 ( z ) + ( 2 1 − d ) log ( 1 − z ) + O ( N − 1 ) , (165) uniformly on compact subsets of C ∖ [ 1 , ∞ ) \mathbb{C}\setminus[1,\infty) C ∖ [ 1 , ∞ ) [77 ] . Applying this expansion to the four q q q (149) yields (164) . The derivative bounds follow from Cauchy’s estimates applied to the analytic relative error on a slightly larger neighbourhood. □
We next construct a contour for the steepest descent analysis. We use the principal branches of the logarithm and dilogarithm; see [86 Section 25.12(i)] . On the negative real axis, Re Φ ( a , z ) \operatorname{Re}\Phi(a,z) Re Φ ( a , z ) [86 Eq. (25.12.4)] . Recall that z ∗ z_* z ∗ (157) .
Lemma 6.8. Assume c 1 > 0 c_1>0 c 1 > 0 c 2 ≥ 0 c_2\geq0 c 2 ≥ 0 Γ s \Gamma_s Γ s
Γ s \Gamma_s Γ s [ − s c 1 + 1 , 0 ] [-s^{c_1+1},0] [ − s c 1 + 1 , 0 ] [ − a , 0 ] [-a,0] [ − a , 0 ]
For some ε > 0 \varepsilon>0 ε > 0 Γ s ∩ { z : ∣ z − z ∗ ∣ < ε } = { z ∗ + t e ± π i / 3 : 0 ≤ t < ε } \Gamma_s\cap\{z:|z-z_*|<\varepsilon\}=\{z_*+te^{\pm\pi i/3}:0\leq t<\varepsilon\} Γ s ∩ { z : ∣ z − z ∗ ∣ < ε } = { z ∗ + t e ± π i /3 : 0 ≤ t < ε }
Re Φ ( a , z ) < Φ ( a , z ∗ ) \operatorname{Re}\Phi(a,z)<\Phi(a,z_*) Re Φ ( a , z ) < Φ ( a , z ∗ ) z ∈ Γ s ∖ { z ∗ } z\in\Gamma_s\setminus\{z_*\} z ∈ Γ s ∖ { z ∗ }
∣ z ∣ ≥ z ∗ |z|\geq z_* ∣ z ∣ ≥ z ∗ z ∈ Γ s z\in\Gamma_s z ∈ Γ s
Proof. We first assume c 2 > 0 c_2>0 c 2 > 0 ∣ z ∣ = z ∗ |z|=z_* ∣ z ∣ = z ∗ (159) ,
z ∗ 2 − s c 1 + c 2 + 1 a = s c 1 + 1 a ( 1 − s c 2 ) 1 − s > 0. z_*^2-s^{c_1+c_2+1}a=\frac{s^{c_1+1}a(1-s^{c_2})}{1-s}>0. z ∗ 2 − s c 1 + c 2 + 1 a = 1 − s s c 1 + 1 a ( 1 − s c 2 ) > 0. Substituting z = z ∗ e i t z=z_*e^{it} z = z ∗ e i t (163) therefore gives Im ϕ ( a , z ∗ e i t ) > 0 \operatorname{Im}\phi(a,z_*e^{it})>0 Im ϕ ( a , z ∗ e i t ) > 0 0 < t < π 0<t<\pi 0 < t < π (156) ,
d d t Re Φ ( a , z ∗ e i t ) = − 1 λ arg ϕ ( a , z ∗ e i t ) < 0. \frac{d}{dt}\operatorname{Re}\Phi(a,z_*e^{it})=-\frac{1}{\lambda}\arg\phi(a,z_*e^{it})<0. d t d Re Φ ( a , z ∗ e i t ) = − λ 1 arg ϕ ( a , z ∗ e i t ) < 0. By continuity and conjugation symmetry, the circle satisfies the third property. Write R = max { s c 1 + 1 , a } R=\max\{s^{c_1+1},a\} R = max { s c 1 + 1 , a } z ∗ > R z_*>R z ∗ > R (159) implies
min { s c 1 + 1 , a } < z ∗ ≤ R < 1 , s c 1 + c 2 + 1 a < z ∗ . \min\{s^{c_1+1},a\}<z_*\le R<1,\qquad s^{c_1+c_2+1}a<z_*. min { s c 1 + 1 , a } < z ∗ ≤ R < 1 , s c 1 + c 2 + 1 a < z ∗ . For z ∗ < r < R z_*<r<R z ∗ < r < R (163) then gives 0 < ϕ ( a , − r ) < s 0<\phi(a,-r)<s 0 < ϕ ( a , − r ) < s
d d r Re Φ ( a , − r ) = 1 λ r log ϕ ( a , − r ) s < 0. \frac{d}{dr}\operatorname{Re}\Phi(a,-r)=\frac{1}{\lambda r}\log\frac{\phi(a,-r)}{s}<0. d r d Re Φ ( a , − r ) = λ r 1 log s ϕ ( a , − r ) < 0. By continuity, a thin outward deformation near the negative real axis can thus extend slightly beyond − R -R − R
Finally, near z ∗ z_* z ∗ z ∗ + t e ± π i / 3 z_*+te^{\pm\pi i/3} z ∗ + t e ± π i /3 0 ≤ t ≤ ε 0\le t\le\varepsilon 0 ≤ t ≤ ε ∣ z − z ∗ ∣ = ε |z-z_*|=\varepsilon ∣ z − z ∗ ∣ = ε z = z ∗ + ε e i t z=z_*+\varepsilon e^{it} z = z ∗ + ε e i t
π 3 ≤ t ≤ t ε , t ε = arccos ( − ε 2 z ∗ ) = π 2 + O ( ε ) . \frac{\pi}{3}\le t\le t_\varepsilon,\qquad t_\varepsilon=\arccos\left(-\frac{\varepsilon}{2z_*}\right)=\frac{\pi}{2}+O(\varepsilon). 3 π ≤ t ≤ t ε , t ε = arccos ( − 2 z ∗ ε ) = 2 π + O ( ε ) . Since Φ z ( a , z ∗ ) = Φ z z ( a , z ∗ ) = 0 \Phi_z(a,z_*)=\Phi_{zz}(a,z_*)=0 Φ z ( a , z ∗ ) = Φ z z ( a , z ∗ ) = 0 Φ z z z ( a , z ∗ ) > 0 \Phi_{zzz}(a,z_*)>0 Φ z z z ( a , z ∗ ) > 0
Re Φ ( a , z ∗ + ε e i t ) − Φ ( a , z ∗ ) = ε 3 6 Φ z z z ( a , z ∗ ) cos ( 3 t ) + O ( ε 4 ) , \operatorname{Re}\Phi(a,z_*+\varepsilon e^{it})-\Phi(a,z_*)=\frac{\varepsilon^3}{6}\Phi_{zzz}(a,z_*)\cos(3t)+O(\varepsilon^4), Re Φ ( a , z ∗ + ε e i t ) − Φ ( a , z ∗ ) = 6 ε 3 Φ z z z ( a , z ∗ ) cos ( 3 t ) + O ( ε 4 ) , d d t Re Φ ( a , z ∗ + ε e i t ) = − ε 3 2 Φ z z z ( a , z ∗ ) sin ( 3 t ) + O ( ε 4 ) . \frac{d}{dt}\operatorname{Re}\Phi(a,z_*+\varepsilon e^{it})=-\frac{\varepsilon^3}{2}\Phi_{zzz}(a,z_*)\sin(3t)+O(\varepsilon^4). d t d Re Φ ( a , z ∗ + ε e i t ) = − 2 ε 3 Φ z z z ( a , z ∗ ) sin ( 3 t ) + O ( ε 4 ) . For a fixed sufficiently small δ > 0 \delta>0 δ > 0 [ π / 3 , π / 2 − δ ] [\pi/3,\pi/2-\delta] [ π /3 , π /2 − δ ] [ π / 2 − δ , t ε ] [\pi/2-\delta,t_\varepsilon] [ π /2 − δ , t ε ] ∣ z ∣ = z ∗ |z|=z_* ∣ z ∣ = z ∗ Φ ( a , z ∗ ) \Phi(a,z_*) Φ ( a , z ∗ ) ∣ z ∣ ≥ z ∗ |z|\ge z_* ∣ z ∣ ≥ z ∗
It remains to consider c 2 = 0 c_2=0 c 2 = 0 (159) gives z ∗ 2 = s c 1 + 1 a z_*^2=s^{c_1+1}a z ∗ 2 = s c 1 + 1 a
0 ≤ ϕ ( a , z ∗ e i t ) = a + s c 1 + 1 + 2 z ∗ cos t 1 + z ∗ 2 + 2 z ∗ cos t < s , 0 < t < 2 π . 0\le\phi(a,z_*e^{it})=\frac{a+s^{c_1+1}+2z_*\cos t}{1+z_*^2+2z_*\cos t}<s,\qquad0<t<2\pi. 0 ≤ ϕ ( a , z ∗ e i t ) = 1 + z ∗ 2 + 2 z ∗ cos t a + s c 1 + 1 + 2 z ∗ cos t < s , 0 < t < 2 π . Thus Re Φ ( a , z ) \operatorname{Re}\Phi(a,z) Re Φ ( a , z ) ∣ z ∣ = z ∗ |z|=z_* ∣ z ∣ = z ∗ (156) shows that a small outward radial deformation away from z ∗ z_* z ∗
To justify exponential localisation near z ∗ z_* z ∗ z ∈ Γ s z\in\Gamma_s z ∈ Γ s
1 N log ∣ ( − q / z ; q ) ∞ ∣ ≤ 1 λ ∫ 0 ∞ log ∣ 1 + e − t / z ∣ d t + o ( 1 ) = − 1 λ Re Li 2 ( − 1 / z ) + o ( 1 ) . \frac{1}{N}\log|(-q/z;q)_\infty|\le\frac{1}{\lambda}\int_0^\infty\log|1+e^{-t}/z|\,dt+o(1)=-\frac{1}{\lambda}\operatorname{Re}\operatorname{Li}_2(-1/z)+o(1). N 1 log ∣ ( − q / z ; q ) ∞ ∣ ≤ λ 1 ∫ 0 ∞ log ∣1 + e − t / z ∣ d t + o ( 1 ) = − λ 1 Re Li 2 ( − 1/ z ) + o ( 1 ) . Indeed, the upper bound follows by taking uniform Riemann sums after replacing the logarithm by
max { log ∣ 1 + e − t / z ∣ , − L } , \max\{\log|1+e^{-t}/z|,-L\}, max { log ∣1 + e − t / z ∣ , − L } , and then letting L → ∞ L\to\infty L → ∞ [86 Eq. (25.12.2)] . For the other three q q q (149) , (165) applies uniformly on the contour for x x x a a a
log ∣ F N ( x , z ) ∣ ≤ N Re Φ ( x , z ) + o ( N ) \log|\mathcal{F}_N(x,z)|\le N\operatorname{Re}\Phi(x,z)+o(N) log ∣ F N ( x , z ) ∣ ≤ N Re Φ ( x , z ) + o ( N ) uniformly there. By compactness and the third property of Lemma 6.8, for any fixed small neighbourhood U U U z ∗ z_* z ∗ M > 0 M>0 M > 0 C o u t > 0 C_{\mathrm{out}}>0 C out > 0
sup u ∈ C ∣ u ∣ ≤ M sup z ∈ Γ s ∖ U ∣ e − N Φ ( x ^ , z ∗ ) F N ( x ^ , z ) ∣ ≤ e − C o u t N (166) \sup_{\substack{u\in\mathbb{C}\\ |u|\le M}}\ \sup_{z\in\Gamma_s\setminus U}\left|e^{-N\Phi(\widehat{x},z_*)}\mathcal{F}_N(\widehat{x},z)\right|\le e^{-C_{\mathrm{out}}N} \tag*{(166)} u ∈ C ∣ u ∣ ≤ M sup z ∈ Γ s ∖ U sup e − N Φ ( x , z ∗ ) F N ( x , z ) ≤ e − C out N (166) for all sufficiently large N N N
By the first property of Lemma 6.8, Γ s \Gamma_s Γ s F ~ N ( x , ⋅ ) \widetilde{\mathcal{F}}_N(x,\cdot) F N ( x , ⋅ ) F ~ N − 1 ( x , ⋅ ) \widetilde{\mathcal{F}}_{N-1}(x,\cdot) F N − 1 ( x , ⋅ ) x x x a a a N N N p N p_N p N p N − 1 p_{N-1} p N − 1
F ~ N − 1 ( x , z ) = F ~ N ( x , z ) R N ( x , z ) , R N ( x , z ) : = z + q N + a + b + 1 x z ( z + q N + a ) . (167) \widetilde{\mathcal{F}}_{N-1}(x,z)=\widetilde{\mathcal{F}}_N(x,z)\mathcal{R}_N(x,z),\qquad\mathcal{R}_N(x,z):=\frac{z+q^{N+a+b+1}x}{z(z+q^{N+a})}. \tag*{(167)} F N − 1 ( x , z ) = F N ( x , z ) R N ( x , z ) , R N ( x , z ) := z ( z + q N + a ) z + q N + a + b + 1 x . (167) Near ( a , z ∗ ) (a,z_*) ( a , z ∗ )
R N ( x , z ) = R ( x , z ) + O ( N − 1 ) , R ( x , z ) : = z + s c 1 + c 2 + 1 x z ( z + s c 1 + 1 ) , \mathcal{R}_N(x,z)=\mathcal{R}(x,z)+O(N^{-1}),\qquad\mathcal{R}(x,z):=\frac{z+s^{c_1+c_2+1}x}{z(z+s^{c_1+1})}, R N ( x , z ) = R ( x , z ) + O ( N − 1 ) , R ( x , z ) := z ( z + s c 1 + 1 ) z + s c 1 + c 2 + 1 x , locally uniformly together with all partial derivatives of any fixed order. Define
q N ( x ) : = p N − 1 ( x ) − R N ( x , z ∗ ) p N ( x ) = ∮ Γ s F ~ N ( x , z ) ( R N ( x , z ) − R N ( x , z ∗ ) ) d z 2 π i z . (168) q_N(x):=p_{N-1}(x)-\mathcal{R}_N(x,z_*)p_N(x)
=\oint_{\Gamma_s}\widetilde{\mathcal{F}}_N(x,z)\bigl(\mathcal{R}_N(x,z)-\mathcal{R}_N(x,z_*)\bigr)\frac{dz}{2\pi i z}. \tag*{(168)} q N ( x ) := p N − 1 ( x ) − R N ( x , z ∗ ) p N ( x ) = ∮ Γ s F N ( x , z ) ( R N ( x , z ) − R N ( x , z ∗ ) ) 2 π i z d z . (168) The Christoffel–Darboux formula (6.28) then becomes
K N ( a , b ) ( x , y ; q ) C N ( a , b ) w ( a , b ) ( x ; q ) w ( a , b ) ( y ; q ) = q N ( x ) p N ( y ) − p N ( x ) q N ( y ) x − y + q N + a + b + 1 z ∗ ( z ∗ + q N + a ) p N ( x ) p N ( y ) , (169) \frac{K_N^{(a,b)}(x,y;q)}{C_N^{(a,b)}\sqrt{w^{(a,b)}(x;q)w^{(a,b)}(y;q)}}=
\frac{q_N(x)p_N(y)-p_N(x)q_N(y)}{x-y}
+\frac{q^{N+a+b+1}}{z_*(z_*+q^{N+a})}p_N(x)p_N(y), \tag*{(169)} C N ( a , b ) w ( a , b ) ( x ; q ) w ( a , b ) ( y ; q ) K N ( a , b ) ( x , y ; q ) = x − y q N ( x ) p N ( y ) − p N ( x ) q N ( y ) + z ∗ ( z ∗ + q N + a ) q N + a + b + 1 p N ( x ) p N ( y ) , (169) with the diagonal understood by continuity.
Lemma 6.9. Under the assumptions of Proposition 6.4, as N → ∞ N\to\infty N → ∞
N 1 / 3 e − N Φ ( x ^ , z ∗ ) p N ( x ^ ) = Ψ ( a , z ∗ ) z ∗ ( 2 Φ z z z ( a , z ∗ ) ) 1 / 3 Ai ( u ) + o ( 1 ) , (170) N^{1/3}e^{-N\Phi(\widehat{x},z_*)}p_N(\widehat{x})=
\frac{\Psi(a,z_*)}{z_*}\left(\frac{2}{\Phi_{zzz}(a,z_*)}\right)^{1/3}\operatorname{Ai}(u)+o(1), \tag*{(170)} N 1/3 e − N Φ ( x , z ∗ ) p N ( x ) = z ∗ Ψ ( a , z ∗ ) ( Φ z z z ( a , z ∗ ) 2 ) 1/3 Ai ( u ) + o ( 1 ) , (170) N 2 / 3 e − N Φ ( x ^ , z ∗ ) q N ( x ^ ) = − Ψ ( a , z ∗ ) R z ( a , z ∗ ) z ∗ ( 2 Φ z z z ( a , z ∗ ) ) 2 / 3 Ai ′ ( u ) + o ( 1 ) . (171) N^{2/3}e^{-N\Phi(\widehat{x},z_*)}q_N(\widehat{x})=
-\frac{\Psi(a,z_*)\mathcal{R}_z(a,z_*)}{z_*}\left(\frac{2}{\Phi_{zzz}(a,z_*)}\right)^{2/3}\operatorname{Ai}'(u)+o(1). \tag*{(171)} N 2/3 e − N Φ ( x , z ∗ ) q N ( x ) = − z ∗ Ψ ( a , z ∗ ) R z ( a , z ∗ ) ( Φ z z z ( a , z ∗ ) 2 ) 2/3 Ai ′ ( u ) + o ( 1 ) . (171) Both error terms and their derivatives of any fixed order tend to zero locally uniformly for u ∈ C u\in\mathbb{C} u ∈ C
Proof. Fix M > 0 M>0 M > 0 U U U z ∗ z_* z ∗ R N \mathcal{R}_N R N Γ s \Gamma_s Γ s Γ s ∖ U \Gamma_s\setminus U Γ s ∖ U ∣ u ∣ ≤ M |u|\le M ∣ u ∣ ≤ M
On the local contour, make the change of variables
z = z ∗ + ( 2 N Φ z z z ( a , z ∗ ) ) 1 / 3 w . z=z_*+\left(\frac{2}{N\Phi_{zzz}(a,z_*)}\right)^{1/3}w. z = z ∗ + ( N Φ z z z ( a , z ∗ ) 2 ) 1/3 w . Lemmas 6.6 and 6.7, in particular (6.37), give
F ~ N ( x ^ , z ) = Ψ ( a , z ∗ ) exp [ N Φ ( x ^ , z ∗ ) + w 3 3 − u w ] ( 1 + O ( N − 1 / 3 ) ) , (172) \widetilde{\mathcal{F}}_N(\widehat{x},z)=\Psi(a,z_*)\exp\left[N\Phi(\widehat{x},z_*)+\frac{w^3}{3}-uw\right]\left(1+O(N^{-1/3})\right), \tag*{(172)} F N ( x , z ) = Ψ ( a , z ∗ ) exp [ N Φ ( x , z ∗ ) + 3 w 3 − u w ] ( 1 + O ( N − 1/3 ) ) , (172) uniformly for ∣ u ∣ ≤ M |u|\le M ∣ u ∣ ≤ M w w w
N 1 / 3 d z z ⟶ ( 2 Φ z z z ( a , z ∗ ) ) 1 / 3 d w z ∗ , N 1 / 3 ( R N ( x ^ , z ) − R N ( x ^ , z ∗ ) ) ⟶ ( 2 Φ z z z ( a , z ∗ ) ) 1 / 3 R z ( a , z ∗ ) w . N^{1/3}\frac{dz}{z}\longrightarrow\left(\frac{2}{\Phi_{zzz}(a,z_*)}\right)^{1/3}\frac{dw}{z_*},\qquad
N^{1/3}\bigl(\mathcal{R}_N(\widehat{x},z)-\mathcal{R}_N(\widehat{x},z_*)\bigr)\longrightarrow
\left(\frac{2}{\Phi_{zzz}(a,z_*)}\right)^{1/3}\mathcal{R}_z(a,z_*)w. N 1/3 z d z ⟶ ( Φ z z z ( a , z ∗ ) 2 ) 1/3 z ∗ d w , N 1/3 ( R N ( x , z ) − R N ( x , z ∗ ) ) ⟶ ( Φ z z z ( a , z ∗ ) 2 ) 1/3 R z ( a , z ∗ ) w . To justify integration over the expanding contour, note that Taylor expansion along the local rays gives
N Re ( Φ ( x ^ , z ) − Φ ( x ^ , z ∗ ) ) ≤ − c ∣ w ∣ 3 + C M ∣ w ∣ , N\operatorname{Re}\bigl(\Phi(\widehat{x},z)-\Phi(\widehat{x},z_*)\bigr)\le-c|w|^3+C_M|w|, N Re ( Φ ( x , z ) − Φ ( x , z ∗ ) ) ≤ − c ∣ w ∣ 3 + C M ∣ w ∣ , for some c , C M > 0 c,C_M>0 c , C M > 0 N N N ∣ u ∣ ≤ M |u|\le M ∣ u ∣ ≤ M x ^ − a = O ( N − 2 / 3 ) \widehat{x}-a=O(N^{-2/3}) x − a = O ( N − 2/3 ) C M ∣ w ∣ C_M|w| C M ∣ w ∣
N 1 / 3 ∣ R N ( x ^ , z ) − R N ( x ^ , z ∗ ) ∣ ≤ C ∣ w ∣ . N^{1/3}\left|\mathcal{R}_N(\widehat{x},z)-\mathcal{R}_N(\widehat{x},z_*)\right|\le C|w|. N 1/3 ∣ R N ( x , z ) − R N ( x , z ∗ ) ∣ ≤ C ∣ w ∣. Thus, by Lemma 6.7, both rescaled integrands admit an integrable majorant of the form
C ( 1 + ∣ w ∣ ) exp ( − c ∣ w ∣ 3 + C M ∣ w ∣ ) . C(1+|w|)\exp\left(-c|w|^3+C_M|w|\right). C ( 1 + ∣ w ∣ ) exp ( − c ∣ w ∣ 3 + C M ∣ w ∣ ) . Dominated convergence therefore reduces the two limits to integrals over the Airy contour Γ A i \Gamma_{\mathrm{Ai}} Γ Ai ∞ e − π i / 3 \infty e^{-\pi i/3} ∞ e − π i /3 0 0 0 ∞ e π i / 3 \infty e^{\pi i/3} ∞ e π i /3 [86 Eq. (9.5.4)] and differentiation under the integral sign,
Ai ( u ) = ∫ Γ A i e w 3 / 3 − u w d w 2 π i , Ai ′ ( u ) = − ∫ Γ A i w e w 3 / 3 − u w d w 2 π i \operatorname{Ai}(u)=\int_{\Gamma_{\mathrm{Ai}}}e^{w^{3}/3-uw}\frac{\mathrm{d}w}{2\pi i}, \qquad\operatorname{Ai}'(u)=-\int_{\Gamma_{\mathrm{Ai}}}we^{w^{3}/3-uw}\frac{\mathrm{d}w}{2\pi i} Ai ( u ) = ∫ Γ Ai e w 3 /3 − u w 2 π i d w , Ai ′ ( u ) = − ∫ Γ Ai w e w 3 /3 − u w 2 π i d w Substitution into (152) and (168) proves the asserted limits locally uniformly for u ∈ C u\in\mathbb{C} u ∈ C u u u
We now complete the proof of Proposition 6.4.
Proof of Proposition 6.4. Write w ( x ; q ) = w ( a , b ) ( x ; q ) w(x;q)=w^{(a,b)}(x;q) w ( x ; q ) = w ( a , b ) ( x ; q ) (165) ,
C N ( a , b ) = N s c 1 − d 1 λ ( 1 − s c 1 + c 2 + 2 ) exp [ − N λ ( Li 2 ( s c 1 + 1 ) + Li 2 ( s c 2 + 1 ) − Li 2 ( s ) − Li 2 ( s c 1 + c 2 + 1 ) − λ 2 ( 1 + c 1 ) ) ] ( 1 − s ) 1 / 2 ( 1 − s c 1 + c 2 + 1 ) d 1 + d 2 + 1 / 2 ( 1 − s c 1 + 1 ) d 1 − 1 / 2 ( 1 − s c 2 + 1 ) d 2 − 1 / 2 ( 1 + O ( N − 1 ) ) . (173) C_N^{(a,b)}=\frac{Ns^{c_1-d_1}}{\lambda(1-s^{c_1+c_2+2})}\exp\left[-\frac{N}{\lambda}\left(\operatorname{Li}_2(s^{c_1+1})+\operatorname{Li}_2(s^{c_2+1})-\operatorname{Li}_2(s)-\operatorname{Li}_2(s^{c_1+c_2+1})-\lambda^2(1+c_1)\right)\right]
\frac{(1-s)^{1/2}(1-s^{c_1+c_2+1})^{d_1+d_2+1/2}}{(1-s^{c_1+1})^{d_1-1/2}(1-s^{c_2+1})^{d_2-1/2}}(1+O(N^{-1})). \tag*{(173)} C N ( a , b ) = λ ( 1 − s c 1 + c 2 + 2 ) N s c 1 − d 1 exp [ − λ N ( Li 2 ( s c 1 + 1 ) + Li 2 ( s c 2 + 1 ) − Li 2 ( s ) − Li 2 ( s c 1 + c 2 + 1 ) − λ 2 ( 1 + c 1 ) ) ] ( 1 − s c 1 + 1 ) d 1 − 1/2 ( 1 − s c 2 + 1 ) d 2 − 1/2 ( 1 − s ) 1/2 ( 1 − s c 1 + c 2 + 1 ) d 1 + d 2 + 1/2 ( 1 + O ( N − 1 )) . (173) Similarly, uniformly for x x x a a a
w ( x ; q ) = x d 1 ( 1 − s c 2 x ) d 2 + 1 / 2 ( 1 − x ) 1 / 2 e − N V ( x ) ( 1 + O ( N − 1 ) ) , (174) w(x;q)=\frac{x^{d_1}(1-s^{c_2}x)^{d_2+1/2}}{(1-x)^{1/2}}e^{-NV(x)}(1+O(N^{-1})), \tag*{(174)} w ( x ; q ) = ( 1 − x ) 1/2 x d 1 ( 1 − s c 2 x ) d 2 + 1/2 e − N V ( x ) ( 1 + O ( N − 1 )) , (174) where V V V (101) . Direct calculation from the definitions and Lemma 6.6 gives
2 Φ x ( a , z ∗ ) = V ′ ( a ) (175) 2\Phi_x(a,z_*)=V'(a) \tag*{(175)} 2 Φ x ( a , z ∗ ) = V ′ ( a ) (175) and the prefactor cancellation
− C N ( a , b ) w ( a ; q ) e 2 N Φ ( a , z ∗ ) N 2 Ψ ( a , z ∗ ) 2 R z ( a , z ∗ ) z ∗ 2 Φ z z z ( a , z ∗ ) = 1 + O ( N − 1 ) . -\frac{C_N^{(a,b)}w(a;q)e^{2N\Phi(a,z_*)}}{N}\frac{2\Psi(a,z_*)^2\mathcal{R}_z(a,z_*)}{z_*^2\Phi_{zzz}(a,z_*)}=1+O(N^{-1}). − N C N ( a , b ) w ( a ; q ) e 2 N Φ ( a , z ∗ ) z ∗ 2 Φ z z z ( a , z ∗ ) 2Ψ ( a , z ∗ ) 2 R z ( a , z ∗ ) = 1 + O ( N − 1 ) . Moreover, (175) and Taylor expansion imply
w ( x ^ ; q ) e 2 N Φ ( x ^ , z ∗ ) w ( a ; q ) e 2 N Φ ( a , z ∗ ) = 1 + O ( N − 1 / 3 ) , \frac{w(\widehat{x};q)e^{2N\Phi(\widehat{x},z_*)}}{w(a;q)e^{2N\Phi(a,z_*)}}=1+O(N^{-1/3}), w ( a ; q ) e 2 N Φ ( a , z ∗ ) w ( x ; q ) e 2 N Φ ( x , z ∗ ) = 1 + O ( N − 1/3 ) , uniformly for bounded real u u u
We now substitute Lemma 6.9 into (169) . Since x ^ − y ^ = − ( u − v ) / ( θ N ) 2 / 3 \widehat{x}-\widehat{y}=-(u-v)/(\theta N)^{2/3} x − y = − ( u − v ) / ( θ N ) 2/3 u = v u=v u = v O ( N − 1 / 3 ) O(N^{-1/3}) O ( N − 1/3 )
1 ( θ N ) 2 / 3 K N ( a , b ) ( x ^ , y ^ ; q ) = K A i ( u , v ) + o ( 1 ) (176) \frac{1}{(\theta N)^{2/3}}K_N^{(a,b)}(\widehat{x},\widehat{y};q)=K_{\mathrm{Ai}}(u,v)+o(1) \tag*{(176)} ( θ N ) 2/3 1 K N ( a , b ) ( x , y ; q ) = K Ai ( u , v ) + o ( 1 ) (176) uniformly for u , v ∈ L M s o f t u,v\in\mathcal{L}^{\mathrm{soft}}_M u , v ∈ L M soft
To pass from local kernel convergence to the Tracy–Widom limit, we apply Johansson’s criterion [64 Lemma 3.1] to the counting-measure kernel K N ( a , b ) K_N^{(a,b)} K N ( a , b ) (139) . The following diagonal estimate provides the required tail control in the void region; its proof is deferred to Appendix § 8 .
Proposition 6.10. Under the assumptions of Proposition 6.4, there exists a constant C > 0 C>0 C > 0
0 ≤ K N ( a , b ) ( k , k ; q ) ≤ C ( θ N ) 2 / 3 ( 1 − q ) q k ( 1 + ( θ N ) 2 / 3 ( a − q k ) ) 2 (177) 0\leq K_N^{(a,b)}(k,k;q)\leq\frac{C(\theta N)^{2/3}(1-q)q^k}{\left(1+(\theta N)^{2/3}(a-q^k)\right)^2} \tag*{(177)} 0 ≤ K N ( a , b ) ( k , k ; q ) ≤ ( 1 + ( θ N ) 2/3 ( a − q k ) ) 2 C ( θ N ) 2/3 ( 1 − q ) q k (177) for all sufficiently large N N N k ∈ Z ≥ 0 k\in\mathbb{Z}_{\geq0} k ∈ Z ≥ 0 q k ≤ a q^k\leq a q k ≤ a
We now combine local kernel convergence with the diagonal estimate to prove Theorem 3.2 (ii).
Proof of Theorem 3.2 (ii). Recall the counting-measure kernel K N ( a , b ) K_N^{(a,b)} K N ( a , b ) (139) . For t ∈ R t \in\mathbb{R} t ∈ R
J N ( t ) = { k : ( θ N ) 2 / 3 ( a − q k ) > t } . (178) J_N(t)=\left\{k:(\theta N)^{2/3}(a-q^k)>t\right\}. \tag*{(178)} J N ( t ) = { k : ( θ N ) 2/3 ( a − q k ) > t } . (178) Note that by definition, we have
P ( x min ≥ a − s ( θ N ) 2 / 3 ) = det ( Id − K N ( a , b ) ) ℓ 2 ( J N ( s ) ) . \mathbb{P}\left(x_{\min}\ge a-\frac{s}{(\theta N)^{2/3}}\right)=\det\left(\operatorname{Id}-K_N^{(a,b)}\right)_{\ell^2(J_N(s))}. P ( x m i n ≥ a − ( θ N ) 2/3 s ) = det ( Id − K N ( a , b ) ) ℓ 2 ( J N ( s )) . Set
ρ N : = N 1 / 3 λ a θ 2 / 3 , k N s o f t ( u ) : = ⌊ − N λ log a + ρ N u ⌋ . \rho_N:=\frac{N^{1/3}}{\lambda a\theta^{2/3}},\qquad k_N^{\mathrm{soft}}(u):=\left\lfloor-\frac{N}{\lambda}\log a+\rho_Nu\right\rfloor. ρ N := λa θ 2/3 N 1/3 , k N soft ( u ) := ⌊ − λ N log a + ρ N u ⌋ . Then
( θ N ) 2 / 3 ( a − q k N s o f t ( u ) ) = u + O ( N − 1 / 3 ) , ρ N ( 1 − q ) ( θ N ) 2 / 3 q k N s o f t ( u ) + k N s o f t ( v ) 2 = 1 + o ( 1 ) . (\theta N)^{2/3}\left(a-q^{k_N^{\mathrm{soft}}(u)}\right)=u+O\left(N^{-1/3}\right),\qquad\rho_N(1-q)(\theta N)^{2/3}q^{\frac{k_N^{\mathrm{soft}}(u)+k_N^{\mathrm{soft}}(v)}{2}}=1+o(1). ( θ N ) 2/3 ( a − q k N soft ( u ) ) = u + O ( N − 1/3 ) , ρ N ( 1 − q ) ( θ N ) 2/3 q 2 k N soft ( u ) + k N soft ( v ) = 1 + o ( 1 ) . locally uniformly for u , v ∈ R u,v\in\mathbb{R} u , v ∈ R (139) and Proposition 6.4 that
ρ N K N ( a , b ) ( k N s o f t ( u ) , k N s o f t ( v ) ; q ) = K A i ( u , v ) + o ( 1 ) (179) \rho_NK_N^{(a,b)}\left(k_N^{\mathrm{soft}}(u),k_N^{\mathrm{soft}}(v);q\right)=K_{\mathrm{Ai}}(u,v)+o(1) \tag*{(179)} ρ N K N ( a , b ) ( k N soft ( u ) , k N soft ( v ) ; q ) = K Ai ( u , v ) + o ( 1 ) (179) locally uniformly. Moreover, K N ( a , b ) K_N^{(a,b)} K N ( a , b ) N N N ( K N ( a , b ) ( k i , k j ; q ) ) i , j = 1 m \left(K_N^{(a,b)}(k_i,k_j;q)\right)_{i,j=1}^m ( K N ( a , b ) ( k i , k j ; q ) ) i , j = 1 m
We next verify the diagonal conditions in [64 Lemma 3.1] . Introduce the scaled lattice coordinates
ξ k : = ( θ N ) 2 / 3 ( a − q k ) , ξ k + 1 − ξ k = ( θ N ) 2 / 3 ( 1 − q ) q k , \xi_k:=(\theta N)^{2/3}(a-q^k),\qquad\xi_{k+1}-\xi_k=(\theta N)^{2/3}(1-q)q^k, ξ k := ( θ N ) 2/3 ( a − q k ) , ξ k + 1 − ξ k = ( θ N ) 2/3 ( 1 − q ) q k , so that J N ( t ) = { k : ξ k > t } J_N(t)=\{k:\xi_k>t\} J N ( t ) = { k : ξ k > t } N 0 N_0 N 0 ( θ N ) 2 / 3 ( 1 − q ) a < 1 (\theta N)^{2/3}(1-q)a<1 ( θ N ) 2/3 ( 1 − q ) a < 1 N ≥ N 0 N\ge N_0 N ≥ N 0 t ≥ 0 t\ge0 t ≥ 0 k ∈ J N ( t ) k\in J_N(t) k ∈ J N ( t ) ξ k > t \xi_k>t ξ k > t ξ k + 1 − ξ k < 1 \xi_{k+1}-\xi_k<1 ξ k + 1 − ξ k < 1
K N ( a , b ) ( k , k ; q ) ≤ C ξ k + 1 − ξ k ( 1 + ξ k ) 2 ≤ 2 C ∫ ξ k ξ k + 1 d r ( 1 + r ) 2 . K_N^{(a,b)}(k,k;q)\le C\frac{\xi_{k+1}-\xi_k}{(1+\xi_k)^2}\le2C\int_{\xi_k}^{\xi_{k+1}}\frac{\mathrm{d}r}{(1+r)^2}. K N ( a , b ) ( k , k ; q ) ≤ C ( 1 + ξ k ) 2 ξ k + 1 − ξ k ≤ 2 C ∫ ξ k ξ k + 1 ( 1 + r ) 2 d r . Summing over k ∈ J N ( t ) k\in J_N(t) k ∈ J N ( t )
sup N ≥ N 0 ∑ k ∈ J N ( t ) K N ( a , b ) ( k , k ; q ) ≤ 2 C 1 + t , t ≥ 0. (180) \sup_{N\ge N_0}\sum_{k\in J_N(t)}K_N^{(a,b)}(k,k;q)\le\frac{2C}{1+t},\qquad t\ge0. \tag*{(180)} N ≥ N 0 sup k ∈ J N ( t ) ∑ K N ( a , b ) ( k , k ; q ) ≤ 1 + t 2 C , t ≥ 0. (180) In particular, the tails vanish uniformly as t → ∞ t\to\infty t → ∞
For fixed M > 0 M>0 M > 0 J N ( − M ) ∖ J N ( 0 ) J_N(-M)\setminus J_N(0) J N ( − M ) ∖ J N ( 0 ) O ( ρ N ) O(\rho_N) O ( ρ N ) (179) bounds the corresponding diagonal entries by O ( ρ N − 1 ) O(\rho_N^{-1}) O ( ρ N − 1 ) (180) at t = 0 t=0 t = 0
sup N ≥ N 0 ∑ k ∈ J N ( − M ) K N ( a , b ) ( k , k ; q ) < ∞ . (181) \sup_{N\ge N_0}\sum_{k\in J_N(-M)}K_N^{(a,b)}(k,k;q)<\infty. \tag*{(181)} N ≥ N 0 sup k ∈ J N ( − M ) ∑ K N ( a , b ) ( k , k ; q ) < ∞. (181) Any finitely many remaining values of N N N ∑ k ≥ 0 K N ( a , b ) ( k , k ; q ) = N \sum_{k\ge0}K_N^{(a,b)}(k,k;q)=N ∑ k ≥ 0 K N ( a , b ) ( k , k ; q ) = N
To relate J N ( t ) J_N(t) J N ( t ) t t t
− N λ log ( a − t ( θ N ) 2 / 3 ) = − N λ log a + ρ N t + O ( N − 1 / 3 ) . -\frac{N}{\lambda}\log\left(a-\frac{t}{(\theta N)^{2/3}}\right)=-\frac{N}{\lambda}\log a+\rho_Nt+O\left(N^{-1/3}\right). − λ N log ( a − ( θ N ) 2/3 t ) = − λ N log a + ρ N t + O ( N − 1/3 ) . Thus the preceding trace bounds also give the required conditions for linear cut-offs; finitely many initial values of N N N F T W F_{\mathrm{TW}} F TW
lim N → ∞ det ( Id − K N ( a , b ) ) ℓ 2 ( J N ( s ) ) = det ( Id − K A i ) L 2 ( s , ∞ ) = F T W ( s ) , \lim_{N\to\infty}\det\left(\operatorname{Id}-K_N^{(a,b)}\right)_{\ell^2(J_N(s))}=\det\left(\operatorname{Id}-K_{\mathrm{Ai}}\right)_{L^2(s,\infty)}=F_{\mathrm{TW}}(s), N → ∞ lim det ( Id − K N ( a , b ) ) ℓ 2 ( J N ( s )) = det ( Id − K Ai ) L 2 ( s , ∞ ) = F TW ( s ) , which completes the proof. ◊
Appendix A. Spectral moments of the q q q
In this appendix, we present a complementary derivation of the explicit candidate for the equilibrium measure in Theorem 3.1 . While this derivation is not needed for the proof given in § 5 , it explains how the explicit form of the limiting density arises naturally. Following the moment method in [30 , 31 , 32 ] , we first derive an explicit formula for the spectral moments of the q q q N N N ρ ( c 1 , c 2 ) \rho^{(c_1,c_2)} ρ ( c 1 , c 2 ) N N N
The central object in the moment method is the spectral moment defined by
m N , p ( a , b ) = E [ ∑ j = 1 N x j p ] (182) m_{N,p}^{(a,b)}=\mathbb{E}\left[\sum_{j=1}^{N}x_j^p\right] \tag*{(182)} m N , p ( a , b ) = E [ j = 1 ∑ N x j p ] (182) where the expectation is taken with respect to (37) . Using the orthogonal polynomial representation of the correlation kernel (128) , the spectral moment can be written as
m N , p ( a , b ) = ∑ j = 0 N − 1 1 h j ( a , b ) ∫ 0 1 x p P j ( a , b ) ( x ; q ) 2 w ( a , b ) ( x ; q ) d q x . (183) m_{N,p}^{(a,b)}=\sum_{j=0}^{N-1}\frac{1}{h_j^{(a,b)}}\int_0^1 x^pP_j^{(a,b)}(x;q)^2w^{(a,b)}(x;q)\,d_qx. \tag*{(183)} m N , p ( a , b ) = j = 0 ∑ N − 1 h j ( a , b ) 1 ∫ 0 1 x p P j ( a , b ) ( x ; q ) 2 w ( a , b ) ( x ; q ) d q x . (183) Recall that the Legendre polynomials { P n ( x ) } n ≥ 0 \{P_n(x)\}_{n\ge0} { P n ( x ) } n ≥ 0
1 1 − 2 x t + t 2 = ∑ p = 0 ∞ t p P p ( x ) , ∣ t ∣ < { 1 ∣ x ∣ + x 2 − 1 , if ∣ x ∣ > 1 , 1 , if ∣ x ∣ ≤ 1 ; (184) \frac{1}{\sqrt{1-2xt+t^2}}=\sum_{p=0}^{\infty}t^pP_p(x),\qquad|t|<
\begin{cases}
\frac{1}{|x|+\sqrt{x^2-1}}, & \text{if } |x|>1,\\
1, & \text{if } |x|\le1;
\end{cases} \tag*{(184)} 1 − 2 x t + t 2 1 = p = 0 ∑ ∞ t p P p ( x ) , ∣ t ∣ < { ∣ x ∣ + x 2 − 1 1 , 1 , if ∣ x ∣ > 1 , if ∣ x ∣ ≤ 1 ; (184) see e.g. [71 Eq. (9.8.68)] . The following proposition gives an explicit formula for the spectral moments, together with their large-N N N
Proposition A.1 (Spectral moments of the q q q Let m N , p ( a , b ) m_{N,p}^{(a,b)} m N , p ( a , b ) q q q
(A) (Closed formula) For any nonnegative integer p p p a , b > − 1 a,b>-1 a , b > − 1
m N , p ( a , b ) = ∑ j = 0 N − 1 ∑ r = 0 j ∧ p q p j + r ( r + a − p ) [ p r ] q 1 − q 2 j + a + b + 1 1 − q 2 j − 2 r + p + a + b + 1 ( q j − r + b + 1 ; q ) r ( q j + a + 1 , q j + a + b + 1 ; q ) p − r ( q j − r + 1 ; q ) r ( q 2 j − r + a + b + 1 ; q ) p − r 2 ( q 2 j − 2 r + p + a + b + 2 ; q ) r . (185) m_{N,p}^{(a,b)}=\sum_{j=0}^{N-1}\sum_{r=0}^{j\wedge p}q^{pj+r(r+a-p)}
\begin{bmatrix}p\\r\end{bmatrix}_q
\frac{1-q^{2j+a+b+1}}{1-q^{2j-2r+p+a+b+1}}
\frac{(q^{j-r+b+1};q)_r(q^{j+a+1},q^{j+a+b+1};q)_{p-r}(q^{j-r+1};q)_r}{(q^{2j-r+a+b+1};q)_{p-r}^2(q^{2j-2r+p+a+b+2};q)_r}. \tag*{(185)} m N , p ( a , b ) = j = 0 ∑ N − 1 r = 0 ∑ j ∧ p q p j + r ( r + a − p ) [ p r ] q 1 − q 2 j − 2 r + p + a + b + 1 1 − q 2 j + a + b + 1 ( q 2 j − r + a + b + 1 ; q ) p − r 2 ( q 2 j − 2 r + p + a + b + 2 ; q ) r ( q j − r + b + 1 ; q ) r ( q j + a + 1 , q j + a + b + 1 ; q ) p − r ( q j − r + 1 ; q ) r . (185) Here,
[ p r ] q : = ( q ; q ) p ( q ; q ) r ( q ; q ) p − r , 0 ≤ r ≤ p . \begin{bmatrix}p\\r\end{bmatrix}_q:=\frac{(q;q)_p}{(q;q)_r(q;q)_{p-r}},\qquad0\le r\le p. [ p r ] q := ( q ; q ) r ( q ; q ) p − r ( q ; q ) p , 0 ≤ r ≤ p . (B) (Large N N N Let a a a b b b (43) , and let q = e − λ / N q=e^{-\lambda/N} q = e − λ / N λ > 0 \lambda>0 λ > 0 p p p
m N , p ( a , b ) = N M p + O ( 1 ) , N → ∞ . (186) m_{N,p}^{(a,b)}=N\mathcal{M}_p+O(1),\qquad N\to\infty. \tag*{(186)} m N , p ( a , b ) = N M p + O ( 1 ) , N → ∞. (186) Here, for c 1 > 0 c_1>0 c 1 > 0
M p : = ∫ 0 1 s t ( 1 − s c 1 ) 1 − s 2 t + c 1 + c 2 P p ( ( 1 − s t + c 1 ) ( 1 − s t + c 1 + c 2 ) + s c 1 ( 1 − s t ) ( 1 − s t + c 2 ) ( 1 − s c 1 ) ( 1 − s 2 t + c 1 + c 2 ) ) p d t , (187) \mathcal{M}_p:=\int_0^1\frac{s^t(1-s^{c_1})}{1-s^{2t+c_1+c_2}}P_p\left(\frac{(1-s^{t+c_1})(1-s^{t+c_1+c_2})+s^{c_1}(1-s^t)(1-s^{t+c_2})}{(1-s^{c_1})(1-s^{2t+c_1+c_2})}\right)^p\,dt, \tag*{(187)} M p := ∫ 0 1 1 − s 2 t + c 1 + c 2 s t ( 1 − s c 1 ) P p ( ( 1 − s c 1 ) ( 1 − s 2 t + c 1 + c 2 ) ( 1 − s t + c 1 ) ( 1 − s t + c 1 + c 2 ) + s c 1 ( 1 − s t ) ( 1 − s t + c 2 ) ) p d t , (187) while for c 1 = 0 c_1=0 c 1 = 0
M p : = ( 2 p p ) ∫ 0 1 ( s t ( 1 − s t ) ( 1 − s t + c 2 ) ( 1 − s 2 t + c 2 ) 2 ) p d t . (188) \mathcal{M}_p:=\binom{2p}{p}\int_0^1\left(\frac{s^t(1-s^t)(1-s^{t+c_2})}{(1-s^{2t+c_2})^2}\right)^p\,dt. \tag*{(188)} M p := ( p 2 p ) ∫ 0 1 ( ( 1 − s 2 t + c 2 ) 2 s t ( 1 − s t ) ( 1 − s t + c 2 ) ) p d t . (188) The derivation of (A.4) takes a different route from the proofs in [30 , 31 , 32 ] , where explicit moment formulas were obtained by combinatorial enumeration based on the Flajolet–Viennot theory. Here, we instead exploit parameter shifts and connection coefficients of orthogonal polynomials, following a related idea used in [78 ] for the classical JUE.
Proof of Proposition A.1 (A). We rewrite (A.2) as
m N , p ( a , b ) = ∑ j = 0 N − 1 1 h j ( a , b ) m p , j ( a , b ) , m p , j ( a , b ) : = ∫ 0 1 x p p j ( a , b ) ( x ; q ) 2 w ( a , b ) ( x ; q ) d q x . (189) m_{N,p}^{(a,b)}=\sum_{j=0}^{N-1}\frac{1}{h_j^{(a,b)}}m_{p,j}^{(a,b)},\qquad m_{p,j}^{(a,b)}:=\int_0^1 x^p p_j^{(a,b)}(x;q)^2w^{(a,b)}(x;q)\,\mathrm{d}_q x. \tag*{(189)} m N , p ( a , b ) = j = 0 ∑ N − 1 h j ( a , b ) 1 m p , j ( a , b ) , m p , j ( a , b ) := ∫ 0 1 x p p j ( a , b ) ( x ; q ) 2 w ( a , b ) ( x ; q ) d q x . (189) By the definition (35) of the little q q q
x p w ( a , b ) ( x ; q ) = w ( a + p , b ) ( x ; q ) . (190) x^p w^{(a,b)}(x;q)=w^{(a+p,b)}(x;q). \tag*{(190)} x p w ( a , b ) ( x ; q ) = w ( a + p , b ) ( x ; q ) . (190) Thus, m p , j ( a , b ) m_{p,j}^{(a,b)} m p , j ( a , b ) ℓ 2 ( w ( a + p , b ) d q x ) \ell^2(w^{(a+p,b)}\,\mathrm{d}_q x) ℓ 2 ( w ( a + p , b ) d q x ) p j ( a , b ) p_j^{(a,b)} p j ( a , b ) { p l ( a + p , b ) } 0 ≤ l ≤ j \{p_l^{(a+p,b)}\}_{0\le l\le j} { p l ( a + p , b ) } 0 ≤ l ≤ j
For this purpose, recall the connection formula for the little q q q [62 Corollary 3.2] :
p n ( a 1 , b 1 ) ( x ; q ) = ∑ k = 0 n c n , k ( a 1 , b 1 ) → ( a 2 , b 2 ) p k ( a 2 , b 2 ) ( x ; q ) . (191) p_n^{(a_1,b_1)}(x;q)=\sum_{k=0}^{n}c_{n,k}^{(a_1,b_1)\to(a_2,b_2)}p_k^{(a_2,b_2)}(x;q). \tag*{(191)} p n ( a 1 , b 1 ) ( x ; q ) = k = 0 ∑ n c n , k ( a 1 , b 1 ) → ( a 2 , b 2 ) p k ( a 2 , b 2 ) ( x ; q ) . (191) where
c n , k ( a 1 , b 1 ) → ( a 2 , b 2 ) = q k ( k − n ) [ n k ] q ( q a 2 + 1 , q n + a 1 + b 1 + 1 ; q ) k ( q a 1 + 1 , q k + a 2 + b 2 + 1 ; q ) k 3 ϕ 2 ( q k − n , q n + k + a 1 + b 1 + 1 , q k + a 2 + 1 q k + a 1 + 1 , q 2 k + a 2 + b 2 + 2 ; q , q ) . (192) c_{n,k}^{(a_1,b_1)\to(a_2,b_2)}
=q^{k(k-n)}\begin{bmatrix}n\\k\end{bmatrix}_q
\frac{(q^{a_2+1},q^{n+a_1+b_1+1};q)_k}{(q^{a_1+1},q^{k+a_2+b_2+1};q)_k}
{}_3\phi_2\left(\begin{matrix}q^{k-n},q^{n+k+a_1+b_1+1},q^{k+a_2+1}\\q^{k+a_1+1},q^{2k+a_2+b_2+2}\end{matrix};q,q\right). \tag*{(192)} c n , k ( a 1 , b 1 ) → ( a 2 , b 2 ) = q k ( k − n ) [ n k ] q ( q a 1 + 1 , q k + a 2 + b 2 + 1 ; q ) k ( q a 2 + 1 , q n + a 1 + b 1 + 1 ; q ) k 3 ϕ 2 ( q k − n , q n + k + a 1 + b 1 + 1 , q k + a 2 + 1 q k + a 1 + 1 , q 2 k + a 2 + b 2 + 2 ; q , q ) . (192) When b 1 = b 2 = b b_1=b_2=b b 1 = b 2 = b 3 ϕ 2 {}_3\phi_2 3 ϕ 2 q q q [71 Eq. (1.11.9)] ,
3 ϕ 2 ( q − n , a , b c , a b c − 1 q 1 − n ; q , q ) = ( a − 1 c , b − 1 c ; q ) n ( c , a − 1 b − 1 c ; q ) n , n = 0 , 1 , 2 , … . (193) {}_3\phi_2\left(\begin{matrix}q^{-n},a,b\\c,abc^{-1}q^{1-n}\end{matrix};q,q\right)
=\frac{(a^{-1}c,b^{-1}c;q)_n}{(c,a^{-1}b^{-1}c;q)_n},\qquad n=0,1,2,\ldots. \tag*{(193)} 3 ϕ 2 ( q − n , a , b c , ab c − 1 q 1 − n ; q , q ) = ( c , a − 1 b − 1 c ; q ) n ( a − 1 c , b − 1 c ; q ) n , n = 0 , 1 , 2 , … . (193) Applying this identity, we obtain
c n , k ( a 1 , b ) → ( a 2 , b ) = q k ( k − n ) [ n k ] q ( q a 2 + 1 , q n + a 1 + b + 1 ; q ) k ( q a 1 + 1 , q k + a 2 + b + 1 ; q ) k ( q − ( n + b ) , q a 1 − a 2 ; q ) n − k ( q k + a 1 + 1 , q − ( n + k + a 2 + b + 1 ) ; q ) n − k . (194) c_{n,k}^{(a_1,b)\to(a_2,b)}
=q^{k(k-n)}\begin{bmatrix}n\\k\end{bmatrix}_q
\frac{(q^{a_2+1},q^{n+a_1+b+1};q)_k}{(q^{a_1+1},q^{k+a_2+b+1};q)_k}
\frac{(q^{-(n+b)},q^{a_1-a_2};q)_{n-k}}{(q^{k+a_1+1},q^{-(n+k+a_2+b+1)};q)_{n-k}}. \tag*{(194)} c n , k ( a 1 , b ) → ( a 2 , b ) = q k ( k − n ) [ n k ] q ( q a 1 + 1 , q k + a 2 + b + 1 ; q ) k ( q a 2 + 1 , q n + a 1 + b + 1 ; q ) k ( q k + a 1 + 1 , q − ( n + k + a 2 + b + 1 ) ; q ) n − k ( q − ( n + b ) , q a 1 − a 2 ; q ) n − k . (194) We now specialise to a 1 = a a_1=a a 1 = a a 2 = a + p a_2=a+p a 2 = a + p
1 h j ( a , b ) m p , j ( a , b ) = ∑ l = 0 j h l ( a + p , b ) h j ( a , b ) ( c j , l ( a , b ) → ( a + p , b ) ) 2 . \frac{1}{h_j^{(a,b)}}m_{p,j}^{(a,b)}
=\sum_{l=0}^{j}\frac{h_l^{(a+p,b)}}{h_j^{(a,b)}}\left(c_{j,l}^{(a,b)\to(a+p,b)}\right)^2. h j ( a , b ) 1 m p , j ( a , b ) = l = 0 ∑ j h j ( a , b ) h l ( a + p , b ) ( c j , l ( a , b ) → ( a + p , b ) ) 2 . Inserting (A.13) and changing variables to l = j − r l=j-r l = j − r
1 h j ( a , b ) m p , j ( a , b ) = ∑ r = 0 j ∧ p h j − r ( a + p , b ) h j ( a , b ) q − 2 r ( j − r ) [ j r ] q 2 ( ( q p + a + 1 , q j + a + b + 1 ; q ) j − r ( q j − r + p + a + b + 1 ; q ) j − r ( q − ( j + b ) , q − p ; q ) r ( q − ( 2 j − r + p + a + b + 1 ) ; q ) r 1 ( q a + 1 ; q ) j ) 2 . \frac{1}{h_j^{(a,b)}}m_{p,j}^{(a,b)}
=\sum_{r=0}^{j\wedge p}\frac{h_{j-r}^{(a+p,b)}}{h_j^{(a,b)}}q^{-2r(j-r)}
\begin{bmatrix}j\\r\end{bmatrix}_q^2
\left(
\frac{(q^{p+a+1},q^{j+a+b+1};q)_{j-r}}{(q^{j-r+p+a+b+1};q)_{j-r}}
\frac{(q^{-(j+b)},q^{-p};q)_r}{(q^{-(2j-r+p+a+b+1)};q)_r}
\frac{1}{(q^{a+1};q)_j}
\right)^2. h j ( a , b ) 1 m p , j ( a , b ) = r = 0 ∑ j ∧ p h j ( a , b ) h j − r ( a + p , b ) q − 2 r ( j − r ) [ j r ] q 2 ( ( q j − r + p + a + b + 1 ; q ) j − r ( q p + a + 1 , q j + a + b + 1 ; q ) j − r ( q − ( 2 j − r + p + a + b + 1 ) ; q ) r ( q − ( j + b ) , q − p ; q ) r ( q a + 1 ; q ) j 1 ) 2 . Here, the restriction r ≤ j r\le j r ≤ j l ≥ 0 l\ge0 l ≥ 0 ( q − p ; q ) r = 0 (q^{-p};q)_r=0 ( q − p ; q ) r = 0 r > p r>p r > p 0 ≤ r ≤ j ∧ p 0\le r\le j\wedge p 0 ≤ r ≤ j ∧ p
( q − n ; q ) k = ( − 1 ) k q − n k + k ( k − 1 ) 2 ( q n − k + 1 ; q ) k , n ∈ R , k ∈ Z ≥ 0 , (q^{-n};q)_k=(-1)^kq^{-nk+\frac{k(k-1)}{2}}(q^{n-k+1};q)_k,\qquad n\in\mathbb{R},\quad k\in\mathbb{Z}_{\ge0}, ( q − n ; q ) k = ( − 1 ) k q − nk + 2 k ( k − 1 ) ( q n − k + 1 ; q ) k , n ∈ R , k ∈ Z ≥ 0 , to the factors with negative exponents yields
( c j , j − r ( a , b ) → ( a + p , b ) ) 2 = q r ( r + 2 a + 1 ) [ j r ] q 2 ( ( q p + a + 1 , q j + a + b + 1 ; q ) j − r ( q j − r + b + 1 , q p − r + 1 ; q ) r ( q j − r + p + a + b + 1 ; q ) j − r ( q 2 j − 2 r + p + a + b + 2 ; q ) r ( q a + 1 ; q ) j ) 2 . (195) \left(c_{j,j-r}^{(a,b)\to(a+p,b)}\right)^2
=q^{r(r+2a+1)}
\begin{bmatrix}j\\r\end{bmatrix}_q^2
\left(
\frac{(q^{p+a+1},q^{j+a+b+1};q)_{j-r}(q^{j-r+b+1},q^{p-r+1};q)_r}
{(q^{j-r+p+a+b+1};q)_{j-r}(q^{2j-2r+p+a+b+2};q)_r(q^{a+1};q)_j}
\right)^2. \tag*{(195)} ( c j , j − r ( a , b ) → ( a + p , b ) ) 2 = q r ( r + 2 a + 1 ) [ j r ] q 2 ( ( q j − r + p + a + b + 1 ; q ) j − r ( q 2 j − 2 r + p + a + b + 2 ; q ) r ( q a + 1 ; q ) j ( q p + a + 1 , q j + a + b + 1 ; q ) j − r ( q j − r + b + 1 , q p − r + 1 ; q ) r ) 2 . (195) On the other hand, the squared norms in (6.2) give
h j − r ( a + p , b ) h j ( a , b ) = q p j − r ( p + a + 1 ) ( q a + 1 ; q ) p ( q a + b + 1 ; q ) p 1 − q 2 j + a + b + 1 1 − q 2 j − 2 r + p + a + b + 1 ( q a + 1 , q a + b + 1 ; q ) j ( q p + a + 1 , q p + a + b + 1 ; q ) j − r ( q j − r + 1 , q j − r + b + 1 ; q ) r . \frac{h_{j-r}^{(a+p,b)}}{h_j^{(a,b)}}=q^{pj-r(p+a+1)}\frac{(q^{a+1};q)_p}{(q^{a+b+1};q)_p}\frac{1-q^{2j+a+b+1}}{1-q^{2j-2r+p+a+b+1}}\frac{(q^{a+1},q^{a+b+1};q)_j}{(q^{p+a+1},q^{p+a+b+1};q)_{j-r}(q^{j-r+1},q^{j-r+b+1};q)_r}. h j ( a , b ) h j − r ( a + p , b ) = q p j − r ( p + a + 1 ) ( q a + b + 1 ; q ) p ( q a + 1 ; q ) p 1 − q 2 j − 2 r + p + a + b + 1 1 − q 2 j + a + b + 1 ( q p + a + 1 , q p + a + b + 1 ; q ) j − r ( q j − r + 1 , q j − r + b + 1 ; q ) r ( q a + 1 , q a + b + 1 ; q ) j . Substituting these expressions, the powers of q q q q p j + r ( r + a − p ) q^{pj+r(r+a-p)} q p j + r ( r + a − p )
( u ; q ) m + n = ( u ; q ) m ( u q m ; q ) n , [ j r ] q 2 ( q p − r + 1 ; q ) r 2 ( q j − r + 1 ; q ) r 2 = [ p r ] q 2 ( q j − r + 1 ; q ) r 2 ( q j − r + 1 ; q ) r 2 , (u;q)_{m+n}=(u;q)_m(uq^m;q)_n,\qquad
\begin{bmatrix}j\\r\end{bmatrix}_q^2\frac{(q^{p-r+1};q)_r^2}{(q^{j-r+1};q)_r^2}
=
\begin{bmatrix}p\\r\end{bmatrix}_q^2\frac{(q^{j-r+1};q)_r^2}{(q^{j-r+1};q)_r^2}, ( u ; q ) m + n = ( u ; q ) m ( u q m ; q ) n , [ j r ] q 2 ( q j − r + 1 ; q ) r 2 ( q p − r + 1 ; q ) r 2 = [ p r ] q 2 ( q j − r + 1 ; q ) r 2 ( q j − r + 1 ; q ) r 2 , and then summing over j = 0 , … , N − 1 j=0,\ldots,N-1 j = 0 , … , N − 1
Next, we turn to the asymptotic analysis of the spectral moment.
Proof of Proposition A.1 (B). The case p = 0 p=0 p = 0 p ≥ 1 p\ge1 p ≥ 1 0 ≤ r ≤ p 0\le r\le p 0 ≤ r ≤ p j ≥ r j\ge r j ≥ r
f r ( a , b ) ( j ) = q p j [ p r ] q 2 1 − q 2 j + a + b + 1 1 − q 2 j − 2 r + p + a + b + 1 A p , j , r ( a , b ) . (196) f_r^{(a,b)}(j)=q^{pj}\begin{bmatrix}p\\r\end{bmatrix}_q^2\frac{1-q^{2j+a+b+1}}{1-q^{2j-2r+p+a+b+1}}A_{p,j,r}^{(a,b)}. \tag*{(196)} f r ( a , b ) ( j ) = q p j [ p r ] q 2 1 − q 2 j − 2 r + p + a + b + 1 1 − q 2 j + a + b + 1 A p , j , r ( a , b ) . (196) where
A p , j , r ( a , b ) : = ( q j − r + b + 1 ; q ) r ( q j + a + 1 , q j + a + b + 1 ; q ) p − r ( q j − r + 1 ; q ) r ( q 2 j − r + a + b + 1 ; q ) p − r ( q 2 j − 2 r + p + a + b + 2 ; q ) r 2 . (197) A_{p,j,r}^{(a,b)}:=\frac{(q^{j-r+b+1};q)_r(q^{j+a+1},q^{j+a+b+1};q)_{p-r}(q^{j-r+1};q)_r}{(q^{2j-r+a+b+1};q)_{p-r}(q^{2j-2r+p+a+b+2};q)_r^2}. \tag*{(197)} A p , j , r ( a , b ) := ( q 2 j − r + a + b + 1 ; q ) p − r ( q 2 j − 2 r + p + a + b + 2 ; q ) r 2 ( q j − r + b + 1 ; q ) r ( q j + a + 1 , q j + a + b + 1 ; q ) p − r ( q j − r + 1 ; q ) r . (197) Set f r ( a , b ) ( j ) = 0 f_r^{(a,b)}(j)=0 f r ( a , b ) ( j ) = 0 j < r j<r j < r
m N , p ( a , b ) = ∑ r = 0 p q r ( r + a − p ) ∑ j = 0 N − 1 f r ( a , b ) ( j ) . (198) m_{N,p}^{(a,b)}=\sum_{r=0}^{p}q^{r(r+a-p)}\sum_{j=0}^{N-1}f_r^{(a,b)}(j). \tag*{(198)} m N , p ( a , b ) = r = 0 ∑ p q r ( r + a − p ) j = 0 ∑ N − 1 f r ( a , b ) ( j ) . (198) Suppose first that c 1 + c 2 > 0 c_1+c_2>0 c 1 + c 2 > 0 t = j / N t=j/N t = j / N
1 − q k = k λ N + O ( N − 2 ) 1-q^k=\frac{k\lambda}{N}+O(N^{-2}) 1 − q k = N k λ + O ( N − 2 ) for each fixed k > 0 k>0 k > 0
q p j [ p r ] q 2 1 − q 2 j + a + b + 1 1 − q 2 j − 2 r + p + a + b + 1 = s p t ( p r ) 2 + O ( N − 1 ) , (199) q^{pj}\begin{bmatrix}p\\r\end{bmatrix}_q^2\frac{1-q^{2j+a+b+1}}{1-q^{2j-2r+p+a+b+1}}=s^{pt}\binom{p}{r}^2+O(N^{-1}), \tag*{(199)} q p j [ p r ] q 2 1 − q 2 j − 2 r + p + a + b + 1 1 − q 2 j + a + b + 1 = s pt ( r p ) 2 + O ( N − 1 ) , (199) uniformly for $r\le j\le N-1.
Next, for a fixed b ∈ R b\in\mathbb{R} b ∈ R c c c
( q a N + b ; q ) c = ∏ k = 0 c − 1 ( 1 − s a q b + k ) = ( 1 − s a ) c + O ( N − 1 ) (q^{aN+b};q)_c=\prod_{k=0}^{c-1}(1-s^a q^{b+k})=(1-s^a)^c+O(N^{-1}) ( q a N + b ; q ) c = k = 0 ∏ c − 1 ( 1 − s a q b + k ) = ( 1 − s a ) c + O ( N − 1 ) holds uniformly for a a a [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) A p , j , r ( a , b ) A_{p,j,r}^{(a,b)} A p , j , r ( a , b ) 1 − s 2 t + c 1 + c 2 ≥ 1 − s c 1 + c 2 > 0 1-s^{2t+c_1+c_2}\ge1-s^{c_1+c_2}>0 1 − s 2 t + c 1 + c 2 ≥ 1 − s c 1 + c 2 > 0
A p , j , r ( a , b ) = ( 1 − s t ) r ( 1 − s t + c 1 ) p − r ( 1 − s t + c 2 ) r ( 1 − s t + c 1 + c 2 ) p − r ( 1 − s 2 t + c 1 + c 2 ) 2 p + O ( N − 1 ) , (200) A_{p,j,r}^{(a,b)}=(1-s^t)^r\frac{(1-s^{t+c_1})^{p-r}(1-s^{t+c_2})^r(1-s^{t+c_1+c_2})^{p-r}}{(1-s^{2t+c_1+c_2})^{2p}}+O(N^{-1}), \tag*{(200)} A p , j , r ( a , b ) = ( 1 − s t ) r ( 1 − s 2 t + c 1 + c 2 ) 2 p ( 1 − s t + c 1 ) p − r ( 1 − s t + c 2 ) r ( 1 − s t + c 1 + c 2 ) p − r + O ( N − 1 ) , (200) uniformly for $r\le j\le N-1.
Combining (A.18) and (A.19), we obtain
f r ( a , b ) ( j ) = F r ( a , b ) ( j / N ) + O ( N − 1 ) , f_r^{(a,b)}(j)=F_r^{(a,b)}(j/N)+O(N^{-1}), f r ( a , b ) ( j ) = F r ( a , b ) ( j / N ) + O ( N − 1 ) , where
F r ( a , b ) ( t ) = s p t ( 1 − s t ) r ( p r ) 2 ( 1 − s t + c 1 ) p − r ( 1 − s t + c 2 ) r ( 1 − s t + c 1 + c 2 ) p − r ( 1 − s 2 t + c 1 + c 2 ) 2 p . F_r^{(a,b)}(t)=s^{pt}(1-s^t)^r\binom{p}{r}^2\frac{(1-s^{t+c_1})^{p-r}(1-s^{t+c_2})^r(1-s^{t+c_1+c_2})^{p-r}}{(1-s^{2t+c_1+c_2})^{2p}}. F r ( a , b ) ( t ) = s pt ( 1 − s t ) r ( r p ) 2 ( 1 − s 2 t + c 1 + c 2 ) 2 p ( 1 − s t + c 1 ) p − r ( 1 − s t + c 2 ) r ( 1 − s t + c 1 + c 2 ) p − r . Thus, summing the uniform error and applying the Riemann sum approximation yields
∑ j = 0 N − 1 F r ( a , b ) ( j ) = N ∫ 0 1 F r ( a , b ) ( t ) d t + O ( 1 ) , \sum_{j=0}^{N-1} F_{r}^{(\mathrm{a,b})}(j)=N\int_{0}^{1}F_{r}^{(\mathrm{a,b})}(t)\,\mathrm{d}t+O(1), j = 0 ∑ N − 1 F r ( a , b ) ( j ) = N ∫ 0 1 F r ( a , b ) ( t ) d t + O ( 1 ) , where the finitely many omitted terms with j < r j<r j < r O ( 1 ) O(1) O ( 1 ) (43) also gives
q r ( r + a − p ) = s c 1 r + O ( N − 1 ) , q^{r(r+a-p)}=s^{c_{1}r}+O(N^{-1}), q r ( r + a − p ) = s c 1 r + O ( N − 1 ) , substitution into (A.17) yields
m N , p ( a , b ) = N ∑ r = 0 p s c 1 r ∫ 0 1 F r ( a , b ) ( t ) d t + O ( 1 ) . (201) m_{N,p}^{(\mathrm{a,b})}=N\sum_{r=0}^{p}s^{c_{1}r}\int_{0}^{1}F_{r}^{(\mathrm{a,b})}(t)\,\mathrm{d}t+O(1). \tag*{(201)} m N , p ( a , b ) = N r = 0 ∑ p s c 1 r ∫ 0 1 F r ( a , b ) ( t ) d t + O ( 1 ) . (201) After simplifying the sum over r r r
∑ r = 0 p ( p r ) 2 x r = ( 1 − x ) p P p ( 1 + x 1 − x ) , \sum_{r=0}^{p}\binom{p}{r}^{2}x^{r}=(1-x)^{p}P_{p}\left(\frac{1+x}{1-x}\right), r = 0 ∑ p ( r p ) 2 x r = ( 1 − x ) p P p ( 1 − x 1 + x ) , the desired asymptotic behaviour (A.5) follows.
For c 1 = c 2 = 0 c_{1}=c_{2}=0 c 1 = c 2 = 0 a = d 1 a=d_{1} a = d 1 b = d 2 b=d_{2} b = d 2 j j j T p , j T_{p,j} T p , j
∑ r = 0 p ( p r ) 2 = ( 2 p p ) , ∑ r = 0 p r ( p r ) 2 = p 2 ( 2 p p ) , \sum_{r=0}^{p}\binom{p}{r}^{2}=\binom{2p}{p},\qquad
\sum_{r=0}^{p}r\binom{p}{r}^{2}=\frac{p}{2}\binom{2p}{p}, r = 0 ∑ p ( r p ) 2 = ( p 2 p ) , r = 0 ∑ p r ( r p ) 2 = 2 p ( p 2 p ) , gives
T p , j = ( 2 p p ) ( q j ( 1 + q j ) 2 ) p + O ( N − 1 + j − 2 ) , T_{p,j}=\binom{2p}{p}\left(\frac{q^{j}}{(1+q^{j})^{2}}\right)^{p}+O(N^{-1}+j^{-2}), T p , j = ( p 2 p ) ( ( 1 + q j ) 2 q j ) p + O ( N − 1 + j − 2 ) , uniformly for J ≤ j < N J\leq j<N J ≤ j < N J J J N N N 1 1 1
We now explain how the limiting moments in Proposition A.1 lead to the explicit candidate equilibrium measure. More precisely, we derive the logarithmic expression for the Cauchy transform in Lemma 5.2 directly from the limiting moments. The boundary-value analysis in § 5 then recovers the density, without the need to repeat the inversion calculation. Throughout, we use the notation B B B R R R (108) and (109) , respectively.
Fix a real number z > 1 z>1 z > 1 [ 0 , 1 ] [0,1] [ 0 , 1 ] Proposition A.1 (B) gives 0 ≤ M p ≤ 1 0\leq\mathcal{M}_{p}\leq1 0 ≤ M p ≤ 1 ∑ p = 0 ∞ M p / z p + 1 \sum_{p=0}^{\infty}\mathcal{M}_{p}/z^{p+1} ∑ p = 0 ∞ M p / z p + 1 c 1 > 0 c_{1}>0 c 1 > 0 c 1 = 0 c_{1}=0 c 1 = 0
∑ p = 0 ∞ M p z p + 1 = 1 z ∑ p = 0 ∞ ∫ 0 1 ( s t ( 1 − s c 1 ) z ( 1 − s 2 t + c 1 + c 2 ) ) p P p ( ( 1 − s t + c 1 ) ( 1 − s t + c 1 + c 2 ) + s c 1 ( 1 − s t ) ( 1 − s t + c 2 ) ( 1 − s c 1 ) ( 1 − s 2 t + c 1 + c 2 ) ) d t = 1 λ z ∑ p = 0 ∞ ∫ s ∞ ( u ( 1 − s c 1 ) z ( 1 − s c 1 + c 2 u 2 ) ) p P p ( ( 1 − s c 1 u ) ( 1 − s c 1 + c 2 u ) + s c 1 ( 1 − u ) ( 1 − s c 2 u ) ( 1 − s c 1 ) ( 1 − s c 1 + c 2 u 2 ) ) d u u , \begin{aligned}
\sum_{p=0}^{\infty}\frac{\mathcal{M}_{p}}{z^{p+1}}
=\frac{1}{z}\sum_{p=0}^{\infty}\int_{0}^{1}\left(\frac{s^{t}(1-s^{c_{1}})}{z(1-s^{2t+c_{1}+c_{2}})}\right)^{p}
P_{p}\left(\frac{(1-s^{t+c_{1}})(1-s^{t+c_{1}+c_{2}})+s^{c_{1}}(1-s^{t})(1-s^{t+c_{2}})}
{(1-s^{c_{1}})(1-s^{2t+c_{1}+c_{2}})}\right)\,\mathrm{d}t \\
&=\frac{1}{\lambda z}\sum_{p=0}^{\infty}\int_{s}^{\infty}\left(\frac{u(1-s^{c_{1}})}{z(1-s^{c_{1}+c_{2}}u^{2})}\right)^{p}
P_{p}\left(\frac{(1-s^{c_{1}}u)(1-s^{c_{1}+c_{2}}u)+s^{c_{1}}(1-u)(1-s^{c_{2}}u)}
{(1-s^{c_{1}})(1-s^{c_{1}+c_{2}}u^{2})}\right)\frac{\mathrm{d}u}{u},
\end{aligned} p = 0 ∑ ∞ z p + 1 M p = z 1 p = 0 ∑ ∞ ∫ 0 1 ( z ( 1 − s 2 t + c 1 + c 2 ) s t ( 1 − s c 1 ) ) p P p ( ( 1 − s c 1 ) ( 1 − s 2 t + c 1 + c 2 ) ( 1 − s t + c 1 ) ( 1 − s t + c 1 + c 2 ) + s c 1 ( 1 − s t ) ( 1 − s t + c 2 ) ) d t = λ z 1 p = 0 ∑ ∞ ∫ s ∞ ( z ( 1 − s c 1 + c 2 u 2 ) u ( 1 − s c 1 ) ) p P p ( ( 1 − s c 1 ) ( 1 − s c 1 + c 2 u 2 ) ( 1 − s c 1 u ) ( 1 − s c 1 + c 2 u ) + s c 1 ( 1 − u ) ( 1 − s c 2 u ) ) u d u , where we used the change of variables u = s t u=s^{t} u = s t v = 1 u + s c 1 + c 2 u v=\frac{1}{u}+s^{c_{1}+c_{2}}u v = u 1 + s c 1 + c 2 u
∑ p = 0 ∞ M p z p + 1 = 1 λ ∫ 1 + s c 1 + c 2 B d v ( z v − 1 − s c 1 ) 2 − 4 s c 1 ( 1 − z ) ( 1 − s c 2 z ) . (202) \sum_{p=0}^{\infty}\frac{\mathcal{M}_{p}}{z^{p+1}}
=\frac{1}{\lambda}\int_{1+s^{c_{1}+c_{2}}}^{B}
\frac{\mathrm{d}v}{\sqrt{(zv-1-s^{c_{1}})^{2}-4s^{c_{1}}(1-z)(1-s^{c_{2}}z)}}. \tag*{(202)} p = 0 ∑ ∞ z p + 1 M p = λ 1 ∫ 1 + s c 1 + c 2 B ( z v − 1 − s c 1 ) 2 − 4 s c 1 ( 1 − z ) ( 1 − s c 2 z ) d v . (202) Evaluating this elementary integral gives
∑ p = 0 ∞ M p z p + 1 = 1 λ z log B z − 1 − s c 1 + R ( z ) 2 ( z − 1 ) = g ( z ) , z > 1 , (203) \sum_{p=0}^{\infty}\frac{\mathcal{M}_{p}}{z^{p+1}}
=\frac{1}{\lambda z}\log\frac{Bz-1-s^{c_{1}}+R(z)}{2(z-1)}
=g(z),\qquad z>1, \tag*{(203)} p = 0 ∑ ∞ z p + 1 M p = λ z 1 log 2 ( z − 1 ) B z − 1 − s c 1 + R ( z ) = g ( z ) , z > 1 , (203) where the last equality follows from Lemma 5.2 .
This calculation derives the logarithmic Cauchy transform (110) directly from the limiting moments. The Sokhotski–Plemelj inversion carried out in the proof of Lemma 5.2 then recovers the density ρ ( c 1 , c 2 ) \rho^{(c_1,c_2)} ρ ( c 1 , c 2 ) (48) , explaining how the explicit candidate equilibrium measure is obtained.
Diagonal estimates at the soft edge
In this appendix, we prove the diagonal estimate, Proposition 6.10. We retain the notation and assumptions of Proposition 6.4, and let Γ s \Gamma_s Γ s Lemma 6.8 . We begin with a uniform bound for the weighted integrand in the void region.
Recall that F ~ N \widetilde{\mathcal{F}}_N F N (149) .
Lemma B.1. Let q = e − λ / N q=e^{-\lambda/N} q = e − λ / N λ > 0 \lambda>0 λ > 0 c v , C v > 0 c_v,C_v>0 c v , C v > 0 N N N
w ( a , b ) ( x ; q ) ∣ F ~ N ( x , z ) ∣ ≤ C v w ( a , b ) ( a ; q ) ∣ F ~ N ( a , z ) ∣ e − c v N ( a − x ) ( ∣ z ∣ − z ∗ ) (204) \sqrt{w^{(a,b)}(x;q)}\left|\widetilde{\mathcal{F}}_N(x,z)\right|\leq C_v\sqrt{w^{(a,b)}(a;q)}\left|\widetilde{\mathcal{F}}_N(a,z)\right|e^{-c_vN(a-x)(|z|-z_*)} \tag*{(204)} w ( a , b ) ( x ; q ) F N ( x , z ) ≤ C v w ( a , b ) ( a ; q ) F N ( a , z ) e − c v N ( a − x ) ( ∣ z ∣ − z ∗ ) (204) uniformly for x ∈ ( 0 , a ] x\in(0,a] x ∈ ( 0 , a ] z ∈ Γ s z\in\Gamma_s z ∈ Γ s z ∗ z_* z ∗ (157) .
Proof. Recall the phase function Φ \Phi Φ (154) and the external potential V V V (101) . Define
g ( x ) : = λ x ( 2 Φ x ( x , z ∗ ) − V ′ ( x ) ) = 2 log z ∗ + s c 1 + c 2 + 1 x z ∗ + x + log 1 − x 1 − s c 2 x + c 1 λ . g(x):=\lambda x\left(2\Phi_x(x,z_*)-V'(x)\right)=2\log\frac{z_*+s^{c_1+c_2+1}x}{z_*+x}+\log\frac{1-x}{1-s^{c_2}x}+c_1\lambda. g ( x ) := λ x ( 2 Φ x ( x , z ∗ ) − V ′ ( x ) ) = 2 log z ∗ + x z ∗ + s c 1 + c 2 + 1 x + log 1 − s c 2 x 1 − x + c 1 λ . This function is strictly decreasing on ( 0 , a ] (0,a] ( 0 , a ] g ( a ) = 0 g(a)=0 g ( a ) = 0 (175) . Thus g ( x ) ≥ 0 g(x)\geq0 g ( x ) ≥ 0 g ( a / 2 ) > 0 g(a/2)>0 g ( a /2 ) > 0
Recall that the contour Γ s \Gamma_s Γ s Lemma 6.8 satisfies ∣ z ∣ ≥ z ∗ |z|\geq z_* ∣ z ∣ ≥ z ∗
Φ x z ( x , r ) = 1 − s c 1 + c 2 + 1 λ ( r + x ) ( r + s c 1 + c 2 + 1 x ) , \Phi_{xz}(x,r)=\frac{1-s^{c_1+c_2+1}}{\lambda(r+x)(r+s^{c_1+c_2+1}x)}, Φ x z ( x , r ) = λ ( r + x ) ( r + s c 1 + c 2 + 1 x ) 1 − s c 1 + c 2 + 1 , there is a constant c Φ > 0 c_\Phi>0 c Φ > 0
2 Φ x ( x , r ) − V ′ ( x ) = g ( x ) λ x + 2 ∫ z ∗ r Φ x z ( x , t ) d t ≥ g ( x ) λ x + c Φ ( r − z ∗ ) (205) 2\Phi_x(x,r)-V'(x)=\frac{g(x)}{\lambda x}+2\int_{z_*}^{r}\Phi_{xz}(x,t)\,dt\geq\frac{g(x)}{\lambda x}+c_\Phi(r-z_*) \tag*{(205)} 2 Φ x ( x , r ) − V ′ ( x ) = λ x g ( x ) + 2 ∫ z ∗ r Φ x z ( x , t ) d t ≥ λ x g ( x ) + c Φ ( r − z ∗ ) (205) uniformly for x ∈ ( 0 , a ] x\in(0,a] x ∈ ( 0 , a ] r ∈ [ z ∗ , max z ∈ Γ s ∣ z ∣ ] r\in[z_*,\max_{z\in\Gamma_s}|z|] r ∈ [ z ∗ , max z ∈ Γ s ∣ z ∣ ] (159) implies z ∗ 2 > s c 1 + c 2 + 1 a 2 z_*^2>s^{c_1+c_2+1}a^2 z ∗ 2 > s c 1 + c 2 + 1 a 2 z ∈ Γ s z\in\Gamma_s z ∈ Γ s r = ∣ z ∣ r=|z| r = ∣ z ∣
∣ exp [ λ x Φ x ( x , z ) ] ∣ = ∣ z + s c 1 + c 2 + 1 x z + x ∣ ≥ r + s c 1 + c 2 + 1 x r + x . (206) \left|\exp[\lambda x\Phi_x(x,z)]\right|=\left|\frac{z+s^{c_1+c_2+1}x}{z+x}\right|\geq\frac{r+s^{c_1+c_2+1}x}{r+x}. \tag*{(206)} ∣ exp [ λ x Φ x ( x , z )] ∣ = z + x z + s c 1 + c 2 + 1 x ≥ r + x r + s c 1 + c 2 + 1 x . (206) Combining these inequalities and integrating from x x x a a a
2 Re Φ ( x , z ) − 2 Re Φ ( a , z ) − ( V ( x ) − V ( a ) ) ≤ − c Φ ( a − x ) ( ∣ z ∣ − z ∗ ) − 1 λ ∫ x a g ( t ) t d t . (207) 2\operatorname{Re}\Phi(x,z)-2\operatorname{Re}\Phi(a,z)-(V(x)-V(a))\leq-c_\Phi(a-x)(|z|-z_*)-\frac{1}{\lambda}\int_x^a\frac{g(t)}{t}\,dt. \tag*{(207)} 2 Re Φ ( x , z ) − 2 Re Φ ( a , z ) − ( V ( x ) − V ( a )) ≤ − c Φ ( a − x ) ( ∣ z ∣ − z ∗ ) − λ 1 ∫ x a t g ( t ) d t . (207) Note that by (149) , cancelling the factors independent of x x x
F ~ N ( x , z ) F ~ N ( a , z ) = ( − q N + a + b + 2 x / z , − q a / z ; q ) ∞ ( − q x / z , − q N + a + b + 2 a / z ; q ) ∞ , \frac{\widetilde{\mathcal{F}}_N(x,z)}{\widetilde{\mathcal{F}}_N(a,z)}
=\frac{(-q^{N+a+b+2}x/z,-qa/z;q)_\infty}{(-qx/z,-q^{N+a+b+2}a/z;q)_\infty}, F N ( a , z ) F N ( x , z ) = ( − q x / z , − q N + a + b + 2 a / z ; q ) ∞ ( − q N + a + b + 2 x / z , − q a / z ; q ) ∞ , which also defines the quotient at common zeros. Since Γ s \Gamma_s Γ s [ − a , 0 ] [-a,0] [ − a , 0 ] C ∖ [ 1 , ∞ ) \mathbb{C}\setminus[1,\infty) C ∖ [ 1 , ∞ ) (35) and (165) that
log ( w ( a , b ) ( x ; q ) w ( a , b ) ( a ; q ) ∣ F ~ N ( x , z ) F ~ N ( a , z ) ∣ ) = N ( Re Φ ( x , z ) − Re Φ ( a , z ) − V ( x ) − V ( a ) 2 ) + d 1 2 log x a + O ( 1 ) , (208) \log\left(\sqrt{\frac{w^{(a,b)}(x;q)}{w^{(a,b)}(a;q)}}\left|\frac{\widetilde{\mathcal{F}}_N(x,z)}{\widetilde{\mathcal{F}}_N(a,z)}\right|\right)
=N\left(\operatorname{Re}\Phi(x,z)-\operatorname{Re}\Phi(a,z)-\frac{V(x)-V(a)}{2}\right)+\frac{d_1}{2}\log\frac{x}{a}+O(1), \tag*{(208)} log ( w ( a , b ) ( a ; q ) w ( a , b ) ( x ; q ) F N ( a , z ) F N ( x , z ) ) = N ( Re Φ ( x , z ) − Re Φ ( a , z ) − 2 V ( x ) − V ( a ) ) + 2 d 1 log a x + O ( 1 ) , (208) uniformly for x ∈ ( 0 , a ] x\in(0,a] x ∈ ( 0 , a ] z ∈ Γ s z\in\Gamma_s z ∈ Γ s
Finally, since g g g
∫ x a g ( t ) t d t ≥ g ( a / 2 ) log a 2 x , 0 < x < a / 2. \int_x^a\frac{g(t)}{t}\,dt\geq g(a/2)\log\frac{a}{2x},\qquad0<x<a/2. ∫ x a t g ( t ) d t ≥ g ( a /2 ) log 2 x a , 0 < x < a /2. It follows that
− N 2 λ ∫ x a g ( t ) t d t + d 1 2 log x a -\frac{N}{2\lambda}\int_x^a \frac{g(t)}{t}\,\mathrm{d}t+\frac{d_1}{2}\log\frac{x}{a} − 2 λ N ∫ x a t g ( t ) d t + 2 d 1 log a x is bounded above uniformly for x ∈ ( 0 , a ] x\in(0,a] x ∈ ( 0 , a ] N N N d 1 < 0 d_1<0 d 1 < 0 c v = c Φ / 2 c_{\mathrm{v}}=c_{\Phi}/2 c v = c Φ /2
We next estimate p N p_N p N q N q_N q N (152) and (168) , together with their derivatives. To remove the common contribution to the derivatives in the Christoffel–Darboux Wronskian, introduce
ℓ N ( x , z ) : = ∂ ∂ x log ( − q N + a + b + 2 x / z ; q ) ∞ ( − q x / z ; q ) ∞ . (209) \ell_N(x,z):=\frac{\partial}{\partial x}\log\frac{(-q^{N+a+b+2}x/z;q)_\infty}{(-qx/z;q)_\infty}. \tag*{(209)} ℓ N ( x , z ) := ∂ x ∂ log ( − q x / z ; q ) ∞ ( − q N + a + b + 2 x / z ; q ) ∞ . (209) By (149) , this satisfies ∂ x F ~ N ( x , z ) = ℓ N ( x , z ) F ~ N ( x , z ) \partial_x\widetilde{F}_N(x,z)=\ell_N(x,z)\widetilde{F}_N(x,z) ∂ x F N ( x , z ) = ℓ N ( x , z ) F N ( x , z )
Lemma B.2. For x ∈ ( 0 , a ] x\in(0,a] x ∈ ( 0 , a ] u = ( θ N ) 2 / 3 ( a − x ) u=(\theta N)^{2/3}(a-x) u = ( θ N ) 2/3 ( a − x ) C > 0 C>0 C > 0 N N N x x x N N N
w ( a , b ) ( x ; q ) ∣ p N ( x ) ∣ ≤ C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) 1 N 1 / 3 ( 1 + u ) (210) \sqrt{w^{(a,b)}(x;q)}|p_N(x)|\le C\sqrt{w^{(a,b)}(a;q)}e^{N\Phi(a,z_*)}\frac{1}{N^{1/3}(1+u)} \tag*{(210)} w ( a , b ) ( x ; q ) ∣ p N ( x ) ∣ ≤ C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) N 1/3 ( 1 + u ) 1 (210) w ( a , b ) ( x ; q ) ∣ q N ( x ) ∣ ≤ C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) 1 N 2 / 3 ( 1 + u ) 2 (211) \sqrt{w^{(a,b)}(x;q)}|q_N(x)|\le C\sqrt{w^{(a,b)}(a;q)}e^{N\Phi(a,z_*)}\frac{1}{N^{2/3}(1+u)^2} \tag*{(211)} w ( a , b ) ( x ; q ) ∣ q N ( x ) ∣ ≤ C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) N 2/3 ( 1 + u ) 2 1 (211) w ( a , b ) ( x ; q ) ∣ p N ′ ( x ) − ℓ N ( x , z ∗ ) p N ( x ) ∣ ≤ C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) N 1 / 3 ( 1 + u ) 2 (212) \sqrt{w^{(a,b)}(x;q)}|p_N'(x)-\ell_N(x,z_*)p_N(x)|\le C\sqrt{w^{(a,b)}(a;q)}e^{N\Phi(a,z_*)}\frac{N^{1/3}}{(1+u)^2} \tag*{(212)} w ( a , b ) ( x ; q ) ∣ p N ′ ( x ) − ℓ N ( x , z ∗ ) p N ( x ) ∣ ≤ C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) ( 1 + u ) 2 N 1/3 (212) w ( a , b ) ( x ; q ) ∣ q N ′ ( x ) − ℓ N ( x , z ∗ ) q N ( x ) ∣ ≤ C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) 1 ( 1 + u ) 3 (213) \sqrt{w^{(a,b)}(x;q)}|q_N'(x)-\ell_N(x,z_*)q_N(x)|\le C\sqrt{w^{(a,b)}(a;q)}e^{N\Phi(a,z_*)}\frac{1}{(1+u)^3} \tag*{(213)} w ( a , b ) ( x ; q ) ∣ q N ′ ( x ) − ℓ N ( x , z ∗ ) q N ( x ) ∣ ≤ C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) ( 1 + u ) 3 1 (213) uniformly for x ∈ ( 0 , a ] x\in(0,a] x ∈ ( 0 , a ]
Proof. We first prove that, for j = 0 , 1 , 2 j=0,1,2 j = 0 , 1 , 2
∫ Γ s ∣ z − z ∗ ∣ j w ( a , b ) ( x ; q ) ∣ F ~ N ( x , z ) z ∣ ∣ d z ∣ ≤ C j w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) N − ( j + 1 ) / 3 ( 1 + u ) − ( j + 1 ) , (214) \int_{\Gamma_s}|z-z_*|^j\sqrt{w^{(a,b)}(x;q)}\left|\frac{\widetilde{F}_N(x,z)}{z}\right|\,|dz|
\le C_j\sqrt{w^{(a,b)}(a;q)}e^{N\Phi(a,z_*)}N^{-(j+1)/3}(1+u)^{-(j+1)}, \tag*{(214)} ∫ Γ s ∣ z − z ∗ ∣ j w ( a , b ) ( x ; q ) z F N ( x , z ) ∣ d z ∣ ≤ C j w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) N − ( j + 1 ) /3 ( 1 + u ) − ( j + 1 ) , (214) uniformly for x ∈ ( 0 , a ] x\in(0,a] x ∈ ( 0 , a ] N N N u = ( θ N ) 2 / 3 ( a − x ) u=(\theta N)^{2/3}(a-x) u = ( θ N ) 2/3 ( a − x )
By Lemma 6.8, the contour near z ∗ z_* z ∗ z = z ∗ + t e ± i π / 3 z=z_*+te^{\pm i\pi/3} z = z ∗ + t e ± iπ /3 0 ≤ t ≤ ε 0\le t\le\varepsilon 0 ≤ t ≤ ε ε \varepsilon ε
Re Φ ( a , z ) − Φ ( a , z ∗ ) ≤ − Φ z z z ( a , z ∗ ) 12 t 3 , ∣ z ∣ − z ∗ ≥ t / 2. \operatorname{Re}\Phi(a,z)-\Phi(a,z_*)\le-\frac{\Phi_{zzz}(a,z_*)}{12}t^3,\qquad|z|-z_*\ge t/2. Re Φ ( a , z ) − Φ ( a , z ∗ ) ≤ − 12 Φ z z z ( a , z ∗ ) t 3 , ∣ z ∣ − z ∗ ≥ t /2. Combining the local expansion (164) with Lemma B.1, we therefore obtain
w ( a , b ) ( x ; q ) ∣ F ~ N ( x , z ) ∣ ≤ C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) e − c N ( t 3 + ( a − x ) t ) \sqrt{w^{(a,b)}(x;q)}|\widetilde{F}_N(x,z)|\le C\sqrt{w^{(a,b)}(a;q)}e^{N\Phi(a,z_*)}e^{-cN(t^3+(a-x)t)} w ( a , b ) ( x ; q ) ∣ F N ( x , z ) ∣ ≤ C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) e − c N ( t 3 + ( a − x ) t ) for some constants C , c > 0 C,c>0 C , c > 0 ∣ z ∣ ≥ z ∗ > 0 |z|\ge z_*>0 ∣ z ∣ ≥ z ∗ > 0 r = N 1 / 3 t r=N^{1/3}t r = N 1/3 t
C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) N − ( j + 1 ) / 3 ∫ 0 ∞ r j e − c ( r 3 + u r ) d r ≤ C j w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) N − ( j + 1 ) / 3 ( 1 + u ) − ( j + 1 ) . C\sqrt{w^{(a,b)}(a;q)}e^{N\Phi(a,z_*)}N^{-(j+1)/3}\int_0^\infty r^j e^{-c(r^3+ur)}\,\mathrm{d}r
\le C_j\sqrt{w^{(a,b)}(a;q)}e^{N\Phi(a,z_*)}N^{-(j+1)/3}(1+u)^{-(j+1)}. C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) N − ( j + 1 ) /3 ∫ 0 ∞ r j e − c ( r 3 + u r ) d r ≤ C j w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) N − ( j + 1 ) /3 ( 1 + u ) − ( j + 1 ) . Here, the constants may be adjusted, and the last inequality follows by considering 0 ≤ u ≤ 1 0\le u\le1 0 ≤ u ≤ 1 u ≥ 1 u\ge1 u ≥ 1
On the remaining part of the contour, (166) , applied at x = a x=a x = a
C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) e − c N . C\sqrt{w^{(a,b)}(a;q)}e^{N\Phi(a,z_*)}e^{-cN}. C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) e − c N . Since 0 ≤ u ≤ a ( θ N ) 2 / 3 0\le u\le a(\theta N)^{2/3} 0 ≤ u ≤ a ( θ N ) 2/3
We now apply it to the contour representations (152) and (168) . Recall ℓ N \ell_N ℓ N R N ( x , z ) \mathcal{R}_N(x,z) R N ( x , z ) (167) . The relevant denominators are uniformly separated from zero for x ∈ [ 0 , a ] x\in[0,a] x ∈ [ 0 , a ] z ∈ Γ s z\in\Gamma_s z ∈ Γ s z z z z ∗ z_* z ∗ ∑ h ≥ 0 q h = ( 1 − q ) − 1 = O ( N ) \sum_{h\geq0}q^h=(1-q)^{-1}=O(N) ∑ h ≥ 0 q h = ( 1 − q ) − 1 = O ( N )
∣ l N ( x , z ) − l N ( x , z ∗ ) ∣ ≤ C N ∣ z − z ∗ ∣ , \left|l_N(x,z)-l_N(x,z_*)\right|\leq CN\left|z-z_*\right|, ∣ l N ( x , z ) − l N ( x , z ∗ ) ∣ ≤ C N ∣ z − z ∗ ∣ , and
∣ R N ( x , z ) − R N ( x , z ∗ ) ∣ ≤ C ∣ z − z ∗ ∣ , ∣ ∂ x R N ( x , z ) − ∂ x R N ( x , z ∗ ) ∣ ≤ C ∣ z − z ∗ ∣ . \left|\mathfrak{R}_N(x,z)-\mathfrak{R}_N(x,z_*)\right|\leq C\left|z-z_*\right|,\qquad\left|\partial_x\mathfrak{R}_N(x,z)-\partial_x\mathfrak{R}_N(x,z_*)\right|\leq C\left|z-z_*\right|. ∣ R N ( x , z ) − R N ( x , z ∗ ) ∣ ≤ C ∣ z − z ∗ ∣ , ∣ ∂ x R N ( x , z ) − ∂ x R N ( x , z ∗ ) ∣ ≤ C ∣ z − z ∗ ∣ . Consequently, the cases j = 0 j=0 j = 0 j = 1 j=1 j = 1
Differentiating the fixed-contour representations and using ∂ x F N = l N F N \partial_x\mathfrak{F}_N=l_N\mathfrak{F}_N ∂ x F N = l N F N
p N ′ ( x ) − l N ( x , z ∗ ) p N ( x ) = ∮ Γ s F ~ N ( x , z ) ( l N ( x , z ) − l N ( x , z ∗ ) ) d z 2 π i z . p_N'(x)-l_N(x,z_*)p_N(x)=\oint_{\Gamma_s}\widetilde{\mathfrak{F}}_N(x,z)\bigl(l_N(x,z)-l_N(x,z_*)\bigr)\frac{dz}{2\pi iz}. p N ′ ( x ) − l N ( x , z ∗ ) p N ( x ) = ∮ Γ s F N ( x , z ) ( l N ( x , z ) − l N ( x , z ∗ ) ) 2 π i z d z . Thus the case j = 1 j=1 j = 1
q N ′ ( x ) − l N ( x , z ∗ ) q N ( x ) = ∮ Γ s F ~ N ( x , z ) ( l N ( x , z ) − l N ( x , z ∗ ) ) ( R N ( x , z ) − R N ( x , z ∗ ) ) d z 2 π i z + ∮ Γ s F ~ N ( x , z ) ( ∂ x R N ( x , z ) − ∂ x R N ( x , z ∗ ) ) d z 2 π i z . \begin{aligned}
q_N'(x)-l_N(x,z_*)q_N(x)
={}&\oint_{\Gamma_s}\widetilde{\mathfrak{F}}_N(x,z)\bigl(l_N(x,z)-l_N(x,z_*)\bigr)\bigl(\mathfrak{R}_N(x,z)-\mathfrak{R}_N(x,z_*)\bigr)\frac{dz}{2\pi iz}\\
&+\oint_{\Gamma_s}\widetilde{\mathfrak{F}}_N(x,z)\bigl(\partial_x\mathfrak{R}_N(x,z)-\partial_x\mathfrak{R}_N(x,z_*)\bigr)\frac{dz}{2\pi iz}.
\end{aligned} q N ′ ( x ) − l N ( x , z ∗ ) q N ( x ) = ∮ Γ s F N ( x , z ) ( l N ( x , z ) − l N ( x , z ∗ ) ) ( R N ( x , z ) − R N ( x , z ∗ ) ) 2 π i z d z + ∮ Γ s F N ( x , z ) ( ∂ x R N ( x , z ) − ∂ x R N ( x , z ∗ ) ) 2 π i z d z . Applying (B.11) with j = 2 j=2 j = 2 j = 1 j=1 j = 1
w ( a , b ) ( x ; q ) ∣ q N ′ ( x ) − l N ( x , z ∗ ) q N ( x ) ∣ ≤ C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) ( ( 1 + u ) − 3 + N − 2 / 3 ( 1 + u ) − 2 ) . \sqrt{w^{(a,b)}(x;q)}\left|q_N'(x)-l_N(x,z_*)q_N(x)\right|\leq C\sqrt{w^{(a,b)}(a;q)}e^{N\Phi(a,z_*)}\left((1+u)^{-3}+N^{-2/3}(1+u)^{-2}\right). w ( a , b ) ( x ; q ) ∣ q N ′ ( x ) − l N ( x , z ∗ ) q N ( x ) ∣ ≤ C w ( a , b ) ( a ; q ) e N Φ ( a , z ∗ ) ( ( 1 + u ) − 3 + N − 2/3 ( 1 + u ) − 2 ) . Finally, N − 2 / 3 ( 1 + u ) ≤ 1 + a θ 2 / 3 N^{-2/3}(1+u)\leq1+a\theta^{2/3} N − 2/3 ( 1 + u ) ≤ 1 + a θ 2/3
We now complete the proof of Proposition 6.10.
Proof of Proposition 6.10. Fix k ∈ Z ≥ 0 k\in\mathbb{Z}_{\geq0} k ∈ Z ≥ 0 q k ≤ a q^k\leq a q k ≤ a x = q k x=q^k x = q k u = ( θ N ) 2 / 3 ( a − x ) ≥ 0 u=(\theta N)^{2/3}(a-x)\geq0 u = ( θ N ) 2/3 ( a − x ) ≥ 0 (169) gives
K N ( a , b ) ( x , x ; q ) C N ( a , b ) w ( a , b ) ( x ; q ) = p N ( x ) q N ′ ( x ) − p N ′ ( x ) q N ( x ) + ∂ x R N ( x , z ∗ ) p N ( x ) 2 . \frac{K_N^{(a,b)}(x,x;q)}{C_N^{(a,b)}w^{(a,b)}(x;q)}=p_N(x)q_N'(x)-p_N'(x)q_N(x)+\partial_x\mathfrak{R}_N(x,z_*)p_N(x)^2. C N ( a , b ) w ( a , b ) ( x ; q ) K N ( a , b ) ( x , x ; q ) = p N ( x ) q N ′ ( x ) − p N ′ ( x ) q N ( x ) + ∂ x R N ( x , z ∗ ) p N ( x ) 2 . The common term involving l N l_N l N
p N ( x ) q N ′ ( x ) − p N ′ ( x ) q N ( x ) = p N ( x ) ( q N ′ ( x ) − l N ( x , z ∗ ) q N ( x ) ) − ( p N ′ ( x ) − l N ( x , z ∗ ) p N ( x ) ) q N ( x ) . p_N(x)q_N'(x)-p_N'(x)q_N(x)=p_N(x)\bigl(q_N'(x)-l_N(x,z_*)q_N(x)\bigr)-\bigl(p_N'(x)-l_N(x,z_*)p_N(x)\bigr)q_N(x). p N ( x ) q N ′ ( x ) − p N ′ ( x ) q N ( x ) = p N ( x ) ( q N ′ ( x ) − l N ( x , z ∗ ) q N ( x ) ) − ( p N ′ ( x ) − l N ( x , z ∗ ) p N ( x ) ) q N ( x ) . Also,
∂ x R N ( x , z ∗ ) = q N + a + b + 1 z ∗ ( z ∗ + q N + a ) = O ( 1 ) , \partial_x\mathfrak{R}_N(x,z_*)=\frac{q^{N+a+b+1}}{z_*(z_*+q^{N+a})}=O(1), ∂ x R N ( x , z ∗ ) = z ∗ ( z ∗ + q N + a ) q N + a + b + 1 = O ( 1 ) , while the prefactor calculation in the proof of Proposition 6.4 gives
C N ( a , b ) w ( a , b ) ( a ; q ) e 2 N Φ ( a , z ∗ ) ≤ C N . C_N^{(a,b)}w^{(a,b)}(a;q)e^{2N\Phi(a,z_*)}\leq CN. C N ( a , b ) w ( a , b ) ( a ; q ) e 2 N Φ ( a , z ∗ ) ≤ C N . Applying Lemma B.2, we therefore obtain
0 ≤ K N ( a , b ) ( x , x ; q ) ( θ N ) 2 / 3 ≤ C ( ( 1 + u ) − 4 + N − 1 / 3 ( 1 + u ) − 2 ) ≤ 2 C ( 1 + u ) − 2 , 0\leq\frac{K_N^{(a,b)}(x,x;q)}{(\theta N)^{2/3}}\leq C\left((1+u)^{-4}+N^{-1/3}(1+u)^{-2}\right)\leq2C(1+u)^{-2}, 0 ≤ ( θ N ) 2/3 K N ( a , b ) ( x , x ; q ) ≤ C ( ( 1 + u ) − 4 + N − 1/3 ( 1 + u ) − 2 ) ≤ 2 C ( 1 + u ) − 2 , uniformly for x = q k ≤ a x=q^k\leq a x = q k ≤ a N N N (177) follows from (139) . □
Acknowledgement
SSB and YGJ were supported by National Research Foundation of Korea grants (RS-2025-00516909, RS-2026-25607520, and RS-2026-25518141). NS gratefully acknowledges support from the Royal Society under grant URF\R\231028. We thank Peter J. Forrester and Kurt Johansson for their interest and helpful comments.
AI use disclosure
The authors declare that generative AI tools (GPT-6 Astra) were used for language polishing and proof checking of the manuscript. All mathematical arguments and proofs were independently developed and written by the authors. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea; School of Mathematics, Korea Institute for Advanced Study, 85 Hoegi-ro, Dongdaemun-gu, Seoul 02455, Republic of Korea
Email address: sungsoobyon@snu.ac.kr
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
Email address: willow21@snu.ac.kr
Department of Mathematics, University of Sussex, Brighton, BN1 9RH, United Kingdom
Email address: n.j.simm@sussex.ac.uk
