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 |$P_{1\cdots k}(s_1,\ldots,s_k)$| of the first to |$k$|th largest/smallest eigenvalues of unitary-invariant random matrices, which is a constituent of the |$k$|-point correlation function |$\rho_k(s_1,\ldots,s_k)$|⁠. 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|${}'$| 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 |$n_1,\ldots,n_N$| in |$\mathfrak{X}$|⁠. (2) Exactly |$k$| particles in |${I}$|⁠, one at each of the |$k$| designated loci |$n_1,\ldots, n_k$| in |${I}$|⁠. (3) |$k$| particles, one at each of the |$k$| designated loci |$n_1,\ldots, n_k$| 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 |$Q_j(x)=((\mathbb{I}-\tilde{\mathbf{K}}_I)^{-1}\cdot x^j {\varphi})(x)$| 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 |$\varphi$| and |$\psi$| such as |$u_j=\int_I dx \,\varphi(x) Q_j(x)$|⁠. 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 |$\Psi(x)$| of an |$\mathbb{R}^2$|-bundle over |$\mathbb{R}$| with an |$\mathfrak{sl}(2,\mathbb{R})$| connection |$\mathcal{A}(x)$|⁠,

  $$ \begin{align} & (\partial_x +\mathcal{A}(x))\Psi(x)=0 ,~~ \Psi(x)= \left[ \begin{array}{l} {\varphi}(x)\\ {\psi}(x) \end{array} \right]\!, \nonumber\\ & \mathcal{A}(x) = -\frac{1}{m(x)} \left[ \begin{array}{rr} {A}(x) & {B}(x)\\ -{C}(x) & -{A}(x) \end{array} \right] ~~\mbox{satisfying}~~ \mathrm{tr}\,\mathcal{A}(x)=0, \end{align}$$

  (25)

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

  $$ \begin{align} & \tilde{\Psi}(x)=\mathcal{U}(x)\Psi(x) ,~~ \tilde{\Psi}(x)= \left[ \begin{array}{l} \tilde{\varphi}(x)\\ \tilde{\psi}(x) \end{array} \right]\!, \nonumber\\ & \mathcal{U}(x)= \left[ \begin{array}{rr} {\displaystyle 1-\frac{a b}{x-t}} & {\displaystyle \frac{b^2}{x-t}} \\ {\displaystyle -\frac{a^2}{x-t}} & {\displaystyle 1+\frac{a b}{x-t}} \end{array} \right] ~~\it{satisfying}~~ \det \mathcal{U}(x)=1. \end{align}$$

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

  $$ \begin{align} & (\partial_x +\tilde{\mathcal{A}}(x))\tilde{\Psi}(x)=0 ,~~ \tilde{\mathcal{A}}(x)=\mathcal{U}(x) \mathcal{A}(x) \mathcal{U}(x)^{-1}- \partial_x \mathcal{U}(x) \cdot \mathcal{U}(x)^{-1} \label{gauge} \end{align}$$

  (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

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

  (28)

  on |${\Psi}(x)$| yields the |$k$|th-order Jánossy density |$J_k(x_1,\ldots,x_k;I)$|⁠. Although the gauge transformation |$\mathcal{U}(x)$| has poles at |$x=x_1,\ldots, x_k$|⁠, the transformed section |$\tilde{\Psi}(x)$| 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 |$J_{k,p}(x_1,\ldots,x_k;I)$| is expressible in terms of a solution to a system of partial differential equations (PDEs) (containing |$x_1,\ldots,x_k$| 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 |$t$|⁠:

  $$ \begin{align} \tilde{K}(x,t)=\tilde{K}(t,y)=0. \label{tildeKxt} \end{align}$$

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

  $$ \begin{align} (\tilde{\mathbf{K}}_I\cdot f)(t)=0\ \ \ \mbox{and thus}\ \ \ ((\mathbb{I}-\tilde{\mathbf{K}}_I)^{-1}\cdot f)(t)=f(t). \label{tildeKft0} \end{align}$$

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

  $$ \begin{align} q_j(t):=((\mathbb{I}-\tilde{\mathbf{K}}_I)^{-1}\cdot x^j \tilde{\varphi})(t)= t^j\tilde{\varphi}(t)=0, \nonumber\\ p_j(t):=((\mathbb{I}-\tilde{\mathbf{K}}_I)^{-1}\cdot x^j\tilde{\psi})(t)= t^j\tilde{\psi}(t)=0 \label{qpt} \end{align}$$

  (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 |$s\gg1$| 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,

  $$ \begin{align} {K}_1(x,y) =\frac{\sqrt{xy}}{2} \frac{J_{3/2}(x)J_{1/2}(y)-J_{1/2}(x)J_{3/2}(y)}{x-y} =\frac{1}{\pi} \left(\frac{\sin(x-y)}{x-y}-\frac{\sin x}{x}\frac{\sin y}{y}\right)\!. \label{K1} \end{align}$$

  (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 |$\mathcal{U}(x)=\left[{~~1~~~\;0\atop-x^{-1} \ 1}\right]$| that retains the tracelessness of |$\mathcal{A}(x)=\left[{0\; -1\atop1~\ \,0}\right].$| 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 (⁠|$k\geq 2$|⁠) of conditioned particles.

4. Applications to random matrix theory

There the functions |$\varphi(x)$| and |$\psi(x)$| are (asymptotic forms of) the |$N$|th and |$(N-1)$|th of the polynomials orthogonal with respect to the weight |$w(x)$|⁠. In this case, the conditional probability distributions |$\tilde{\rho}_p(x_1,\ldots,x_p; t_1,\ldots,t_k)$| (8) and |$\tilde{J}_{k,p}(t_1,\ldots,t_k; I)$| (12) described by the transformed kernel |$\tilde{\mathbf{K}}$| are nothing other than unconditional probability distributions of eigenvalues of a random matrix ensemble with weight function |$\tilde{w}(x)=w(x)\prod_{j=1}^k (x-t_j)^2$|⁠, and |$\tilde{\varphi}(x)$| and |$\tilde{\psi}(x)$| are (asymptotic forms of) polynomials orthogonal with respect to |$\tilde{w}(x)$|⁠.¹

If the values of the resolvent kernel |$R(x,y)=\left\langle x|\mathbf{K}_{I}(\mathbb{I}-\mathbf{K}_{I})^{-1}| y\right\rangle$| for arbitrary |$x, y\in I$| (not just its boundary values |$R(a_i, a_j)$| at |$a_i,a_j \in \partial I$|⁠, as derived in TW) were analytically available for various kernels appearing in the RMT, the Jánossy densities would readily be computed from the first line of Eq. (14). Since this path is infeasible (despite the fact that numerical evaluation of |$\left\langle x|\mathbf{K}_{I}(\mathbb{I}-\mathbf{K}_{I})^{-1}| y\right\rangle$| or |$\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}_{I})$| by the quadrature approximation is always possible [22]), we choose the second line of Eq. (14) as an alternate route. As illustrative examples, we compute Jánossy densities for the Airy and Bessel kernels by applying the TW method to their transformed kernels.

4.1. Jánossy density for the Airy kernel

As an example we concentrate on the simplest of Jánossy densities, |$J_{1}(t; I)$| with |$I=(s,\infty)$|⁠, and already set |${z}$| to unity. Note that |${P}_{12}(t,s)=\Theta(t-s)\partial_s J_{1}(t; (s,\infty))$| represents the joint distribution of the first and second largest eigenvalues |$(t,s)$| of unitary ensembles, previously derived in Ref. [13] via Ref. [12] using a much more elaborate method than this work.

For numerical evaluation of the solution, in practice we impose the boundary condition |$q_0(\Lambda)= \varphi(\Lambda)$|⁠, etc., at a sufficiently large positive |$\Lambda$||$(\sim 10)$|⁠. Since |$q_0(s)$| and |$p_0(s)$| are regular at |$s=t$| (they are actually zero by Eq. (31)), apparent “double poles” at |$s=t$| in the first two nonuniversal equations of Eq. (39) are guaranteed to be canceled by the double zeroes on the right-hand side. Nevertheless, this could potentially cause loss of numerical accuracy when solving the TW system of ODEs from |$s=\Lambda$| down to |$s<t$|⁠, e.g., by the explicit Runge–Kutta method. We have verified that this apparent stiffness at |$s=t$| can be circumvented by adding to |$t$| a tiny imaginary part |$\epsilon$| of the order of |$O(10^{-10})$|⁠. With appropriately chosen values of |$\epsilon=\Im m(t)$|⁠, the real parts of |$q_0(s)$| and |$p_0(s)$| are stable upon varying |$\epsilon$|⁠, and |$q_0(\Re e(t))$| and |$p_0(\Re e(t))$| vanish up to the accuracy of |$O(\epsilon)$| as they should. In Fig. 4 we display the joint distribution of the largest eigenvalue |$t$| and the second largest eigenvalue |$s$|

Fig. 4.

The joint distribution of the first and second largest eigenvalues |${P}_{12}(t,s)$| (orange) and the two-point correlation function |${\rho}_2(t, s)$| for |$t>s$| (transparent blue) of random Hermitian matrices.

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4.2. Jánossy density for the Bessel kernel

Fig. 5.

The joint distribution of the first and second largest singular values |${P}_{12}(t,s)$| (orange) and the two-point correlation function |${\rho}_2(t, s)$| for |$t>s$| (transparent blue) of random complex matrices with |$\nu=0$| (left) and |$\nu=1$| (right).

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5. Conclusion and perspectives

In this article we have shown that the TW method is applicable to the evaluation of Jánossy densities and joint eigenvalue distributions for a kernel |$\mathbf{K}\doteq (\varphi(x)\psi(y)-\psi(x)\varphi(y))/(x-y)$| if it is applicable to the gap probability. Essential to the inheritance of the TW criteria from |$\mathbf{K}$| to the transformed kernel |$\tilde{\mathbf{K}}\doteq (\tilde{\varphi}(x)\tilde{\psi}(y)-\tilde{\psi}(x)\tilde{\varphi}(y))/(x-y)$| is the structure that the map between the component functions |$(\varphi, \psi)\mapsto (\tilde{\varphi}, \tilde{\psi})$| is an |$\mathrm{SL}(2,\mathbb{R})$| gauge transformation to a covariantly constant section of an |$\mathbb{R}^2$|-bundle with an |$\mathfrak{sl}(2,\mathbb{R})$| connection

We list the pros and cons of our approach. In contrast to the model-specific approaches in the preceding works [12,13,15,16], which computed only the first-order Jánossy densities for the Bessel, Airy, and sine kernels, our method is universally and systematically applicable to any kernel satisfying the TW criteria, including but not limited to |$q$|-orthogonal, beyond-Airy, and various finite-|$N$| kernels, and to any |$k$|th-order Jánossy densities. In exchange, the intrinsic connection between our formulation and the isomonodromic systems associated with Painlevé transcendents and integrability in these works is completely obscured. Our approach is not well suited for asymptotic analysis for |$|t-s|\gg 1$| or |$|t-s|\ll 1$|⁠, either.

Finally we comment on possible extensions and physical applications of our approach.

• The joint distribution |$P_{1\cdots k}(s_1,\ldots,s_{k})$| of the first |$k$| extremal eigenvalues is trivially obtained by repeating the procedure (19) |$(k-1)$| times, which increases the order of the polynomials |$\tilde{A}(x)$|⁠, etc. by |$2(k-1)$|⁠. The joint distribution |$P_{p_1\cdots p_\ell}(s_{p_1},\ldots,s_{p_\ell})$| of the |$p_1\mathrm{th}, \ldots, p_\ell\mathrm{th}$| extremal eigenvalues follows from |$P_{1\cdots k}(s_1,\ldots,s_{k})$| by integrating out |$k-\ell$| eigenvalues in an ordered cell, such as |$P_{13}(s_1,s_3)=\int_{s_1}^{s_3}ds_2\,P_{123}(s_1,s_2,s_3)$|⁠.

• For the applicability of the TW method to inherit from |$\mathbf{K}$| to |$\tilde{\mathbf{K}}$|⁠, the requirement of the Christoffel–Darboux form (17), characteristic of |$\mathrm{U}(N)$| invariant ensembles, can actually be relaxed to more generic, asymmetric kernels of the integrable class [25]:

  $$ \begin{align} K(x,y)=\sum_{\ell=1}^r\frac{f_\ell(x) g_\ell(y)}{x-y}:=\frac{{\boldsymbol{\mathit{f}}}(x)\cdot {\boldsymbol{\mathit{g}}}(y)}{x-y} ,~~ {\boldsymbol{\mathit{f}}}(x)\cdot{\boldsymbol{\mathit{g}}}(x)=0. \label{Kr} \end{align}$$

  (52)
  Here |$r$|-component real functions |${\boldsymbol{\mathit{f}}}(x)=\left(f_1(x),\ldots,f_r(x)\right)^t$| and |${\boldsymbol{\mathit{g}}}(y)=\left(g_1(y),\ldots,g_r(y)\right)^t$| are covariantly constant sections for some meromorphic |$\mathfrak{sl}(r,\mathbb{R})$| connections |$\mathcal{A}(x)$| and |$\mathcal{B}(y)$|⁠, respectively. In this generalized case, an |$\mathrm{SL}(r,\mathbb{R})$| gauge transformation on them,

  $$ \begin{align} \begin{array}{ll} {\displaystyle {\boldsymbol{\mathit{f}}}(x)\mapsto \tilde{{\boldsymbol{\mathit{f}}}}(x)={\boldsymbol{\mathit{f}}}(x)-\frac{K(x,t)}{K(t,t)}{\boldsymbol{\mathit{f}}}(t)=\mathcal{U}(x){\boldsymbol{\mathit{f}}}(x)} \\ {\displaystyle {\boldsymbol{\mathit{g}}}(y)\mapsto \tilde{{\boldsymbol{\mathit{g}}}}(y)={\boldsymbol{\mathit{g}}}(y)-\frac{K(t,y)}{K(t,t)}{\boldsymbol{\mathit{g}}}(t)=\mathcal{U}(y)^{-1\,t}{\boldsymbol{\mathit{g}}}(y)} \end{array} ,~~ \mathcal{U}(x)=\mathbb{I}-\frac{{\boldsymbol{\mathit{f}}}(t){\boldsymbol{\mathit{g}}}(t)^t}{\rho_1(t)(x-t)} \end{align}$$

  (53)

  maps |$K(x,y)$| to |$\tilde{K}(x,y)=\tilde{{\boldsymbol{\mathit{f}}}}(x)\cdot\tilde{{\boldsymbol{\mathit{g}}}}(y)/(x-y)$| while retaining |$\tilde{{\boldsymbol{\mathit{f}}}}(x)\cdot\tilde{{\boldsymbol{\mathit{g}}}}(x)=0.$| An example of a kernel of type (52) is the Pearcey kernel (with |$r=3$|⁠) governing spectral correlations of random matrices in an external source, |$H=H_{\mathrm{GUE}}+c\,\mathrm{diag}(\mathbb{I}_{N/2}, -\mathbb{I}_{N/2})$| in the critical regime where a gap in the eigenvalue support closes at the origin [26]. This ensemble schematically models the QCD Dirac operator at finite temperature [27]. Application of our strategy to its Jánossy density by the generalized TW method [28] will be reported in a separate publication.

• Ensembles of Dirichlet |$L$|-functions are acknowledged as ideal quantum-chaotic systems for their distributions of zeroes on the critical line [29,30]. It is well anticipated but worth verifying that the joint distributions of the two smallest zeroes of |$L$|-functions are described by Jánossy densities for the Bessel kernels (51) at |$\nu=\pm 1/2$|⁠, depending on the sign in the functional equations of the |$L$|-functions.

• In the context of noncritical string theory, conditioning the loci of some (⁠|$k$|⁠) of |$N$| eigenvalues of matrix models at (multi)criticality (i.e., beyond Airy) outside their main support has been interpreted as introducing |$k$| ZZ branes to the Liouville theory [31,32]. As all efforts have been concentrated on extracting leading nonperturbative corrections to the free energy in the large-|$N$| limit, it is worthwhile to apply our analytic strategy for computing the Jánossy density |$J_k(\{x\};I)$| to those models and obtain unapproximated, fully nonperturbative free energy that incorporates all D-brane contributions.

• It seems less promising to extend our strategy to Jánossy densities of quaternion kernels [11] governing orthogonal and symplectic ensembles [33], or transitive ensembles interpolating different symmetry classes. Nevertheless, the observation that the Jánossy density for these cases is expressed as a Fredholm Pfaffian of the transformed quaternion kernel [34,35], |$\mathrm{Det}\left(\mathbb{I}-(\mathbf{K}-{\boldsymbol{\mathit{k}}}^t {\boldsymbol\kappa}^{-1} {\boldsymbol{\mathit{k}}})_I\right)^{1/2}$|⁠, always permits numerical evaluation by the quadrature approximation. Currently we are exploring the application of this strategy to the quaternion kernel of the chGSE–chGUE transitive ensemble [36], to obtain individual distributions of the staggered Dirac operator of two-color QCD at finite baryon-number chemical potential |$\mu$| and with dynamical quarks of masses |$m_f$| introduced as the conditioned eigenvalues |$x_f=-m_f^2$|⁠, extending our previous work [6] on the quenched case.

Acknowledgements

I thank Peter Forrester for helpful comments on the manuscript. This work is supported in part by a Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (C) No. 7K05416.

Supplementary material

Numerical data of the Fredholm determinant |$\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}_{(s,\infty)})$| for the Airy kernel in the range |$-7\leq s,t \leq 5$| are attached as |$\texttt{FDAiry.dat}$|⁠. Numerical data of the Fredholm determinant |$\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}_{(0,s)})$| (after replacements |$t \mapsto t^2, s\mapsto s^2$|⁠) for the Bessel kernels at |$\nu=0$| and |$\nu=1$| in the range |$0\leq t,s \leq 9$| are attached as |$\texttt{FDBessel0.dat}$| and |$\texttt{FDBessel1.dat}$|⁠.

Footnotes

References