Exponential Moments and Piecewise Thinning for the Bessel Point Process

Abstract

We obtain exponential moment asymptotics for the Bessel point process. As a direct consequence, we improve on the asymptotics for the expectation and variance of the associated counting function and establish several central limit theorems. We show that exponential moment asymptotics can also be interpreted as large gap asymptotics, in the case where we apply the operation of a piecewise constant thinning on several consecutive intervals. We believe our results also provide important estimates for later studies of the global rigidity of the Bessel point process.

Global rigidity and exponential moments for soft and hard edge point processes

“…An important question in recent years in random matrix theory has been to understand how much the ordered eigenvalues of a random matrix can deviate from their typical locations. It has been observed [Johansson 1998;Gustavsson 2005;Arguin et al 2017;Erdős et al 2009;2012;Claeys et al 2019a] that the individual eigenvalues fluctuate on scales that are only slightly bigger than the microscopic scale. This property is loosely called the rigidity of random matrix eigenvalues.…”

“…Similar optimal global rigidity results have been obtained in circular β-ensembles [Chhaibi et al One of the most important features of random matrix eigenvalues is their universal nature: their asymptotic behavior on microscopic scales is similar for large classes of random matrix models. For instance, in many matrix models of Hermitian n × n matrices, like the GUE, Wigner matrices, and unitary invariant matrices, the microscopic large n behavior of bulk eigenvalues is described by the sine point process (see, e.g., [Erdős and Yau 2017]), whereas the microscopic behavior of edge eigenvalues is described by the Airy point process [Deift and Gioev 2007;Deift et al 1999;Bourgade et al 2014;Forrester 1993;Prähofer and Spohn 2002;Soshnikov 2000;Tracy and Widom 1994a]. For ensembles of positive-definite Hermitian matrices, the situation is somewhat more complicated.…”

“…In the Wishart-Laguerre ensemble and its unitary invariant generalizations, the Bessel point process typically describes the microscopic behavior of the smallest eigenvalues close to the hard edge 0 [Forrester 1993;Kuijlaars and Vanlessen 2002;Tracy and Widom 1994b]. However, in a generalization of the Wishart-Laguerre ensemble known as the Muttalib-Borodin Laguerre ensemble [Borodin 1999;Muttalib 1995], the local behavior of eigenvalues near the hard edge is described by a different determinantal point process known as Wright's generalized Bessel point process [Borodin 1999;Kuijlaars and Molag 2019]. Another generalization of the Bessel process, known as the Meijer-G point process, arises at the hard edge of Wishart-type products of Ginibre or truncated unitary matrices [Akemann et al 2013a;2013b;Kuijlaars and Zhang 2014;Kuijlaars and Stivigny 2014;, and in Cauchy multimatrix ensembles [Bertola and Bothner 2015;Bertola et al 2014].…”

“…We do this by combining asymptotics for exponential moments of the counting measures of these point processes, which are Fredholm determinants of certain integral operators, with a global rigidity estimate which can be applied to general point processes which almost surely have a smallest (or largest) point. In the case of the Airy and Bessel point processes, asymptotics for the exponential moments are known (see [Bothner and Buckingham 2018;Charlier and Claeys 2020] for Airy and [Bothner et al 2019;Charlier 2020] for Bessel) and they allow us to improve on the best known upper bounds for the Airy point process (see [Zhong 2019] and [Corwin and Ghosal 2020, Theorem 1.6]), and to deduce completely new global rigidity results for the Bessel point process. As global rigidity estimates give an intuitive idea of how the particles in a point process behave, we believe that they may lead to a better understanding of these random matrix point processes.…”

“…Estimates for the first exponential moment however do not allow us to obtain a sharp lower bound for the global rigidity. For this, one would need more delicate information, like estimates for higher exponential moments, about more complicated averages in the point process, or about convergence of the counting function to a Gaussian multiplicative chaos measure; see, e.g., [Arguin et al 2017;Berestycki et al 2018;Claeys et al 2019a;. In point processes arising in random matrix theory for which optimal lower bounds for the global rigidity are available (see, e.g., [Arguin et al 2017;Chhaibi et al 2018;Claeys et al 2019a;Holcomb and Paquette 2018;Paquette and Zeitouni 2018]), it turns out that the upper bounds obtained via the first exponential moment are sharp, and therefore we believe that the upper bound in Theorem 1.2 is, at least for the concrete examples considered below related to random matrix theory, close to optimal.…”