Asymptotic forms for hard and soft edge general β conditional gap probabilities

Abstract

An infinite log-gas formalism, due to Dyson, and independently Fogler and Shklovskii, is applied to the computation of conditioned gap probabilities at the hard and soft edges of random matrix β-ensembles. The conditioning is that there are n eigenvalues in the gap, with $n\ll |t|$, t denoting the end point of the gap. It is found that the entropy term in the formalism must be replaced by a term involving the potential drop to obtain results consist with known asymptotic expansions in the case $n=0$. With this modification made for general n, the derived expansions — which are for the logarithm of the gap probabilities — are conjectured to be correct up to and including terms $\mathrm{O}(\log |t|)$. They are shown to satisfy various consistency conditions, including an asymptotic duality formula relating β to $4/\beta$.

Introduction

Fundamental to Dysonʼs [13] pioneering studies of random matrices is the log-gas picture. The essential point is that the joint eigenvalue probability density function for various random matrix ensembles can be computed exactly, and is seen to be identical to the Boltzmann factor for certain classical particle systems interacting via a repulsive logarithmic pair potential. These so-called log-gas systems are in equilibrium at particular values of the dimensionless inverse temperature β, restricted to the value $\beta =1$ in the case of the matrix ensemble exhibiting an orthogonal symmetry, $\beta =2$ for a unitary symmetry and $\beta =4$ for a unitary symplectic symmetry. In the application of random matrix ensembles to quantum mechanics these cases correspond to a time reversal symmetry T with $T^2=1$, no time reversal symmetry, and a time reversal symmetry T with $T^2=-1$, respectively.

The significance of the log-gas picture is that it permits the use of macroscopic arguments to compute certain probabilistic quantities in the bulk scaling limit (where the number of eigenvalues goes to infinity with their bulk density fixed to the value ρ). Specifically, the particle system is approximated as a continuum fluid confined to a one-dimensional domain but obeying the laws of two-dimensional electrostatics. As an example, consider $E_{\beta }^{\mathrm{bulk}}(0;(0,t))$, the probability that there are no eigenvalues in the interval $(0,t)$ for the bulk scaling limit of a random matrix ensemble indexed by β. Dysonʼs first application of the log-gas analogy was to derive the asymptotic form

$$\log E_{\beta }^{\mathrm{bulk}}(0;(0,t))\underset{t\to \infty }{\sim }-\beta \frac{{(\pi \rho t)}^2}{16}+(\frac{\beta }{2}-1)\frac{\pi \rho t}{2}+\mathrm{O}(\log \rho t).$$

Although it was remarked [13, above Eq. (92)] that the variational principle underlying (1.1) may contain errors of order $\log \rho t$, other arguments were presented that suggested the log-gas prediction of the explicit term

$$\frac{{(1-\beta /2)}^2}{2\beta }\log \rho t,$$

may in fact be correct at this order.

A noteworthy feature is that (1.1) is valid for all $\beta >0$, and thus in particular applies uniformly to all three random matrix symmetries. In fact, the general $\beta >0$ form of (1.1) is relevant to contemporary random matrix theory [23]. Thus there are random tridiagonal matrices [11] and random Hessenberg unitary matrices [24] that β-generalize the eigenvalue probability density functions for the Gaussian and circular ensembles from classical random matrix theory. A remarkable development relating to these constructions has been the characterization of $E_{\beta }^{\mathrm{bulk}}(0;(0,t))$ in terms of a stochastic differential equation [25], [33]. This was subsequently used [34] to give a rigorous derivation of (1.1), and to furthermore establish that the explicit form of the $\mathrm{O}(\log \rho t)$ term is not (1.2) but rather

$$\frac{1}{4}(\frac{\beta }{2}+\frac{2}{\beta }-3)\log \rho t.$$

Generalizing $E_{\beta }^{\mathrm{bulk}}(0;(0,t))$ is the probability $E_{\beta }^{\mathrm{bulk}}(n;(0,t))$ of there being exactly n eigenvalues in the interval $(0,t)$ for the bulk scaling limit of a random matrix ensemble indexed by β. Many years after his pioneering work [13], Dyson [14] and independently Fogler and Shklovskii [18] used the log-gas picture — now applied directly to the infinite bulk state (in [13] use was made of a scaling limit of a large deviation formula in the finite system; see Section 5) — to derive the generalization of (1.1)

$$\log E_{\beta }^{\mathrm{bulk}}(n;(0,t))\underset{\stackrel{t,n\to \infty }{t\gg n}}{\sim }-\beta \frac{{(\pi \rho t)}^2}{16}+(\beta n+\frac{\beta }{2}-1)\frac{\pi \rho t}{2}+\frac{n}{2}(1-\frac{\beta }{2}-\frac{\beta n}{2})(\log \frac{4\pi \rho t}{n}+1).$$

In fact, in the cases $\beta =1,2$ and 4 this same formula, except requiring n to be fixed, had been obtained earlier [2] using an analysis based on explicit Painlevé/Fredholm determinant expression for $E_{\beta }^{\mathrm{bulk}}(0;(0,t))$.

The primary aim of the present paper is to extend Dysonʼs [14] log-gas study of the bulk conditioned gap probability $E_{\beta }^{\mathrm{bulk}}(n;(0,t))$ to the analogous quantities $E_{\beta }^{\mathrm{soft}}(n;(t,\infty ))$ and $E_{\beta }^{\mathrm{hard}}(n;(0,t))$ corresponding to there being exactly n eigenvalues in the intervals $(t,\infty )$ and $(0,t)$ about the soft and hard spectrum edges respectively, for the regime $1\ll n\ll |t|$, $|t|\to \infty$. The soft edge refers to the neighbourhood of the largest or smallest eigenvalue, in the situation that the corresponding eigenvalue density exhibits a square root profile, which from our formalism extending [14] is taken to be (semi-)infinite in extent. The hard edge is the neighbourhood of the smallest eigenvalue for matrices with non-negative eigenvalues, when the spectral density exhibits an inverse square root singularity profile, which again is to be taken of (semi-)infinite extent.

As examples of explicit realizations of these edge states in terms of limits of finite N matrix ensembles, let ${\text{ME}}_{\beta ,N}(g)$ denote the eigenvalue probability density proportional to

$$\prod _{l=1}^{N}g({\lambda }_l)\prod _{1⩽j<k⩽N}{|{\lambda }_k-{\lambda }_j|}^{\beta }.$$

The choice $g(\lambda )=e^{-\beta {\lambda }^2/2}$ defines the Gaussian β-ensemble, and the choice $g(\lambda )={\lambda }^{\beta a/2}{e}^{-\beta \lambda /2}$ ($\lambda >0$) defines the Laguerre β-ensemble. We know from [19] that the largest eigenvalues are, respectively, to leading order in N at $\lambda =\sqrt{2N}$ and $\lambda =4N$, and that [16], [28] upon the scalings [19]

$${\lambda }_l=\sqrt{2N}+\frac{x_l}{\sqrt{2}{N}^{1/6}},{\lambda }_l=4N+2{(2N)}^{1/2}{x}_l,$$

the correlations have the same well defined $N\to \infty$. The corresponding scaled density ${\rho }_{(1)}^{\mathrm{soft},\beta }(x)$ admits an explicit β-dimensional integral formula in the case of β even [10], and this exhibits the square root singularity characterizing the soft edge state by way of the asymptotic formula

$${\rho }_{(1)}^{\mathrm{soft},\beta }(x)\underset{x\to -\infty }{\sim }\frac{\sqrt{|x|}}{\pi }.$$

The standard realization of the hard edge is the Laguerre β-ensemble, upon the scaling [15], [19]

$${\lambda }_l=\frac{x_l}{4N}.$$

For $a=c-2/\beta$, c a positive integer, and β even the corresponding scaled density ${\rho }_{(1)}^{\mathrm{hard},\beta }(x)$ admits an explicit β-dimensional integral formula [20]. Asymptotic analysis of this integral shows

$${\rho }_{(1)}^{\mathrm{hard},\beta }(x)\underset{x\to \infty }{\sim }\frac{1}{2\pi \sqrt{x}},$$

thus exhibiting the characterizing density profile at the hard edge.

The relevance of the spectrum edge to problems in theoretical and mathematical physics was not apparent in the era of Dysonʼs first works. In fact, at the soft edge there is a somewhat hidden, but nonetheless fundamental, relationship to growth problems of the KPZ class (see e.g. the recent review [17]). And in the case of the hard edge the statistical properties of the smallest eigenvalue can be related to data from lattice QCD simulations (see e.g. [32]).

A priori the log-gas formalism can be applied to the study of the asymptotic form of gap probabilities in the cases the gap contains no eigenvalues, or in the cases there are n eigenvalues with $1\ll n\ll |t|$. In the former case the asymptotic form of the logarithm of the gap probability for general β at both the hard and soft edges is known from other considerations [20], [5], [4], [29], up to and including terms $\mathrm{O}(\log |t|)$. This information is important for two reasons. First, it tells us the infinite log-gas formalism at both the hard and soft edge is in error in the second order (free energy/entropy) contribution. But fortunately, guided by the bulk case, an ansatz can be made to correct this error. Second, it allows the asymptotic expansions obtained for $1\ll n\ll |t|$ to be very naturally extended to hold (as a conjecture) uniformly in n, up to and including terms $\mathrm{O}(\log |t|)$.

These latter expansions, which are the main result of the paper, read

$$\log E_{\beta }^{\mathrm{hard}}(n;(0,t))\underset{\stackrel{t\to \infty }{n\ll t}}{\sim }-\beta \{\frac{t}{8}-\sqrt{t}(n+\frac{a}{2})+[\frac{n^2}{2}+\frac{na}{2}+\frac{a(a-1)}{4}+\frac{a}{2\beta }]\log t^{1/2}\},$$

and

$$\log E_{\beta }^{\mathrm{soft}}(n;(t,\infty ))\underset{\stackrel{t\to -\infty }{n\ll |t|}}{\sim }-\frac{\beta {|t|}^3}{24}+\frac{2\sqrt{2}}{3}{|t|}^{3/2}(\beta n+\frac{\beta }{2}-1)+[\frac{\beta }{2}{n}^2+(\frac{\beta }{2}-1)n+\frac{1}{6}(1-\frac{2}{\beta }{(1-\frac{\beta }{2})}^2)]\log {|t|}^{-3/4}.$$

As a check, using results from [22], asymptotic duality equations — relating gap probabilities with given $\beta ,n$ to gap probabilities at $4/\beta$ and with modified n and length scale — are identified for both $E_{\beta }^{\mathrm{hard}}(n;(0,t))$ and $E_{\beta }^{\mathrm{soft}}(n;(t,\infty ))$, and (1.7), (1.8) are shown to exhibit the required inter-relations.