Tracy–Widom method for Jánossy density and joint distribution of extremal eigenvalues of random matrices

Abstract

1. Introduction

Fig. 1.

Distributions $P_k(s)$ of the scaled $k$th largest eigenvalues of random Hermitian matrices (Tracy–Widom distribution) (right) and of the $k$th smallest singular values of random complex square matrices (left) (red (⁠$k=1$⁠) to blue (⁠$k=8$⁠)), their sums $\sum _{k=1}^8{P}_k(s)$ (grey dotted), and the spectral densities ${\rho }_1(s)$ (black).

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With this in view, the purpose of this article is to advance the formula (1) a step further and provide a “user-friendly” analytic method to compute the joint distribution $P_{1\cdots k}(s_1,\dots ,s_k)$ of the first to $k$th largest/smallest eigenvalues of unitary-invariant random matrices, which is a constituent of the $k$-point correlation function ${\rho }_k(s_1,\dots ,s_k)$⁠. To this end we apply the strategy of Tracy and Widom [7] on the evaluation of Fredholm determinants of integrable integral kernels to the Jánossy density [8–11], i.e., the probability distribution that an interval contains no eigenvalue except for those at $k$ designated loci. As the simplest examples we shall evaluate the joint distributions $P_{12}(t,s)$ of the first and second largest eigenvalues and smallest singular values (see Fig. 2 for their histograms), i.e., the first peak that constitutes the two-point correlation function ${\rho }_2(t,s)=\sum _{k<\ell }{P}_{k\ell }(t,s)$⁠, for the Airy and Bessel kernels. Each case has been worked out previously in Refs. [12,13], which devised an elaborate analytic procedure involving the Painlevé II and III${}^{\prime }$ transcendents and the associated isomonodromic systems [14]. This approach is later simplified (for the Airy kernel) using a solution to the Lax pair associated with the Painlevé XXXIV system [15]. Our alternative method presented in this article, which does not explicitly refer to these systems and employs the familiar Tracy–Widom method, has a clear advantage of permitting straightforward generalizations to a general $P_{1\cdots k}$ and/or to various finite-$N$ and large-$N$ kernels (Hermite, Laguerre, and other hypergeometric; circular, beyond-Airy, $q$-orthogonal, etc.) appearing in the RMT.

Fig. 2.

Histograms of the first $(t)$ and second $(s)$ largest eigenvalues of random Hermitian matrices (left) and of the first $(t)$ and second $(s)$ smallest singular values of random complex square matrices (right). Matrix rank $N=128$ and number of samples $={10}^7$ for each case, and eigen/singular values $x$ are rescaled as: $t$ or $s=\sqrt{2}{N}^{1/6}(x-\sqrt{2N})$ and $\sqrt{2N}x$⁠, respectively.

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This article is composed of the following parts: In Sect. 2 we list known formulas on Jánossy densities of a determinantal point process, and then express them in terms of Fredholm determinants of the “transformed kernel” $\tilde{\mathbf{K}}$⁠. The latter is a novel presentation to the best of our knowledge, except for the simplest (⁠$k=1$⁠) case of the sine kernel previously treated in Ref. [16]. In Sect. 3 we demonstrate that $\tilde{\mathbf{K}}$ satisfies Tracy and Widom’s criteria for their functional-analytic method to be applicable if the original $\mathbf{K}$ does. In Sect. 4 we evaluate Jánossy densities and joint distributions of the first and second extremal eigenvalues from the Airy and Bessel kernels by the Tracy–Widom method. In Sect. 5 we conclude with listing possible applications and extensions of our approach. Numerical data of Jánossy densities for the Airy kernel and the Bessel kernels at $\nu =0,1$ are attached as supplementary material. Throughout this article we follow the notations of Ref. [7], hereafter denoted as TW.

2. Jánossy density

Fig. 3.

Distribution of particles in a DPP. (1) $N$ particles distributed exclusively on $N$ loci $n_1,\dots ,n_N$ in $\mathfrak{X}$⁠. (2) Exactly $k$ particles in $I$⁠, one at each of the $k$ designated loci $n_1,\dots ,n_k$ in $I$⁠. (3) $k$ particles, one at each of the $k$ designated loci $n_1,\dots ,n_k$ and other exactly $p$ particles on $p$ undesignated loci in $I$⁠.

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3. Applicability of the Tracy–Widom method

3.1. Inheritance of the Tracy–Widom criteria

The tracelessness of the $2\times 2$ matrix on the right-hand side of Eq. (18) is essential. As a unifying approach to their preceding works on the sine [18], Airy [3], and Bessel kernels [4], Tracy and Widom have shown in TW that the Fredholm determinant $\mathrm{Det}(\mathbb{I}-{\mathbf{K}}_I)$ of an operator $\mathbf{K}$ satisfying the criteria (17), (18) is always determined through a closed system of PDEs in the boundary points $\{a_i\}\in \partial I$⁠. This involves the boundary values of the functions $Q_j(x)=((\mathbb{I}-{\tilde{\mathbf{K}}}_I{)}^{-1}\cdot x^j\phi )(x)$ and $P_j(x)=((\mathbb{I}-{\tilde{\mathbf{K}}}_I{)}^{-1}\cdot x^j\psi )(x)$⁠, and the inner products of $Q_j$ and $P_j$ with $\phi$ and $\psi$ such as $u_j={\int }_{I}dx\phi (x)Q_j(x)$⁠. A large part of the TW system (Eqs. (1.7a)–(1.9) and (2.12)–(2.18) of TW) is universal and the rest (Eqs. (2.25), (2.26) of TW) parametrically depends on the coefficients of the polynomials $m(x)=\sum _j{\mu }_j{x}^j$⁠, $A(x)=\sum _j{\alpha }_j{x}^j$⁠, etc.

Now we present a theorem:

3.2. Conditioning particles’ loci as gauge transformation

Below we unravel the origin of inheritance of the TW criteria (17) and (18) from $\mathbf{K}$ to $\tilde{\mathbf{K}}$⁠.

• (i)

  The Christoffel–Darboux form (17): suppose that $\mathbf{K}$ is composed of polynomials orthogonal with respect to a weight $w(x)$ or their asymptotic limits. Then $\tilde{\mathbf{K}}$ is composed of polynomials orthogonal with respect to a weight $\tilde{w}(x)=(x-t{)}^{2}w(x)$ or their asymptotic limits, with the factor $(x-t{)}^2$ originating from the Vandermonde determinant squared.

• (ii)
  The tracelessness of the $2\times 2$ matrix in Eq. (18): Eq. (18) specifies a covariantly constant section $\mathrm{\Psi }(x)$ of an ${\mathbb{R}}^2$-bundle over $\mathbb{R}$ with an $\mathfrak{sl}(2,\mathbb{R})$ connection $\mathcal{A}(x)$⁠,

  $$\begin{array}{rl}& ({\partial }_x+\mathcal{A}(x))\mathrm{\Psi }(x)=0,\mathrm{\Psi }(x)=\left[\begin{array}{c}\phi (x)\\ \psi (x)\end{array}\right],\\ & \mathcal{A}(x)=-\frac{1}{m(x)}\left[\begin{array}{rr}A(x)& B(x)\\ -C(x)& -A(x)\end{array}\right]\text{satisfying}\mathrm{tr}\mathcal{A}(x)=0,\end{array}$$

  (25)

  and Eq. (22) is an $\mathrm{SL}(2,\mathbb{R})$ gauge transformation,

  $$\begin{array}{rl}& \tilde{\mathrm{\Psi }}(x)=\mathcal{U}(x)\mathrm{\Psi }(x),\tilde{\mathrm{\Psi }}(x)=\left[\begin{array}{c}\tilde{\phi }(x)\\ \tilde{\psi }(x)\end{array}\right],\\ & \mathcal{U}(x)=\left[\begin{array}{rr}{\displaystyle 1-\frac{ab}{x-t}}& {\displaystyle \frac{b^2}{x-t}}\\ {\displaystyle -\frac{a^2}{x-t}}& {\displaystyle 1+\frac{ab}{x-t}}\end{array}\right]\mathit{s}\mathit{a}\mathit{t}\mathit{i}\mathit{s}\mathit{f}\mathit{y}\mathit{i}\mathit{n}\mathit{g}det\mathcal{U}\mathit{(}\mathit{x}\mathit{)}\mathit{=}\mathit{1.}\end{array}$$

  (26)
  Then the gauge-transformed section $\tilde{\mathrm{\Psi }}(x)$ must be covariantly constant

  $$\begin{array}{rl}& ({\partial }_x+\tilde{\mathcal{A}}(x))\tilde{\mathrm{\Psi }}(x)=0,\tilde{\mathcal{A}}(x)=\mathcal{U}(x)\mathcal{A}(x)\mathcal{U}(x{)}^{-1}-{\partial }_x\mathcal{U}(x)\cdot \mathcal{U}(x{)}^{-1}\end{array}$$

  (27)

  for the gauge-transformed $\mathfrak{sl}(2,\mathbb{R})$ connection $\tilde{A}(x)$that remains traceless, $\mathrm{tr}\tilde{\mathcal{A}}(x)=0$⁠. Repetition of gauge transformations of the form

  $$\begin{array}{r}\mathcal{U}(x)=\left[\begin{array}{rr}{\displaystyle 1-\frac{a_k{b}_k}{x-x_k}}& {\displaystyle \frac{b_k^2}{x-x_k}}\\ {\displaystyle -\frac{a_k^2}{x-x_k}}& {\displaystyle 1+\frac{a_k{b}_k}{x-x_k}}\end{array}\right]\cdots \left[\begin{array}{rr}{\displaystyle 1-\frac{a_1{b}_1}{x-x_1}}& {\displaystyle \frac{b_1^2}{x-x_1}}\\ {\displaystyle -\frac{a_1^2}{x-x_1}}& {\displaystyle 1+\frac{a_1{b}_1}{x-x_1}}\end{array}\right]\end{array}$$

  (28)

  on $\mathrm{\Psi }(x)$ yields the $k$th-order Jánossy density $J_k(x_1,\dots ,x_k;I)$⁠. Although the gauge transformation $\mathcal{U}(x)$ has poles at $x=x_1,\dots ,x_k$⁠, the transformed section $\tilde{\mathrm{\Psi }}(x)$ is regular and vanishes there.
• (iii)

  Meromorphy of $\mathcal{A}(x)$ inherits down to $\tilde{\mathcal{A}}(x)$ by Eq. (27) (which is equivalent to Eq. (24)), as $\mathcal{U}(x)$ is meromorphic.

Accordingly, the TW method is applicable to the evaluation of Jánossy densities of any continuous DPP if it is applicable to the evaluation of its gap probability, and $J_{k,p}(x_1,\dots ,x_k;I)$ is expressible in terms of a solution to a system of partial differential equations (PDEs) (containing $x_1,\dots ,x_k$ parametrically) in the endpoints $\{a_i\}$ of $I$⁠.

A few comments are in order:

• By construction (20), the transformed kernel $\tilde{K}(x,y)$ vanishes when one of the arguments is equal to $t$⁠:

  $$\begin{array}{r}\tilde{K}(x,t)=\tilde{K}(t,y)=0.\end{array}$$

  (29)
  This leads to, for ${}^{\mathrm{\forall }}f\in L^2(I)$⁠,

  $$\begin{array}{r}({\tilde{\mathbf{K}}}_I\cdot f)(t)=0\text{and thus}((\mathbb{I}-{\tilde{\mathbf{K}}}_I{)}^{-1}\cdot f)(t)=f(t).\end{array}$$

  (30)
• As mentioned above, the functions $\tilde{\phi }(x)$ and $\tilde{\psi }(x)$ also vanish at $x=t$⁠. This leads to

  $$\begin{array}{r}{q}_j(t):=((\mathbb{I}-{\tilde{\mathbf{K}}}_I{)}^{-1}\cdot x^j\tilde{\phi })(t)=t^j\tilde{\phi }(t)=0,\\ p_j(t):=((\mathbb{I}-{\tilde{\mathbf{K}}}_I{)}^{-1}\cdot x^j\tilde{\psi })(t)=t^j\tilde{\psi }(t)=0\end{array}$$

  (31)

  for $j\in \mathbb{N}$⁠. These could serve as part of the boundary conditions for the TW system, but we later use them only for a consistency check of the solution $q_j(s)$ and $p_j(s)$ derived from a different boundary condition imposed either at $s\gg 1$ or $s\ll 1$⁠.
• For the sine kernel ${\displaystyle K(x,y)=\frac{\sin (x-y)}{\pi (x-y)}=\frac{\sqrt{xy}}{2}\frac{J_{1/2}(x)J_{-1/2}(y)-J_{-1/2}(x)J_{1/2}(y)}{x-y}}$ governing the spectral bulk of unitary ensembles, Forrester and Odlyzko [16] previously considered the Fredholm determinant of its transformed kernel [19] (which they denoted as $K_1$ instead of our $\tilde{K}$⁠) with $k=1$ and $t$ set to 0 without loss of generality,

  $$\begin{array}{r}{K}_1(x,y)=\frac{\sqrt{xy}}{2}\frac{J_{3/2}(x)J_{1/2}(y)-J_{1/2}(x)J_{3/2}(y)}{x-y}=\frac{1}{\pi }(\frac{\sin (x-y)}{x-y}-\frac{\sin x}{x}\frac{\sin y}{y}).\end{array}$$

  (32)

  They did apply the TW method to $K_1(x,y)$ and expressed the Fredholm determinant on a symmetric interval $\mathrm{Det}(\mathbb{I}-{\mathbf{K}}_1{|}_{(-s,s)})$ in terms of a solution to the TW system of ordinary differential equations (ODEs). However, in order to invoke the TW method they paid attention to an apparent fact that $K_1$ is related to $K$ by a unit shift of the indices of the Bessel functions, rather than by an $\mathrm{SL}(2,\mathbb{R})$ gauge transformation $\mathcal{U}(x)=\left[\genfrac{}{}{0ex}{}{10}{-x^{-1}1}\right]$ that retains the tracelessness of $\mathcal{A}(x)=\left[\genfrac{}{}{0ex}{}{0-1}{10}\right].$ Nor did they explicitly write $K_1$ in a form of the right-hand side of Eq. (32), which would have meant $K(x,y)-K(x,0)K(0,0{)}^{-1}K(0,y)$⁠. Our formulation is a systematic generalization of the spirit of their work to arbitrary kernels of the TW type, to any interval $I$⁠, and to any number (⁠$k\ge 2$⁠) of conditioned particles.