Operator level limit of the circular Jacobi $β$-ensemble

Abstract

We prove an operator level limit for the circular Jacobi β-ensemble. As a result, we characterize the counting function of the limit point process via coupled systems of stochastic differential equations. We also show that the normalized characteristic polynomials converge to a random analytic function, which we characterize via the joint distribution of its Taylor coefficients at zero and as the solution of a stochastic differential equation system. We also provide analogous results for the real orthogonal β-ensemble.

Convergence and an explicit formula for the joint moments of the Circular Jacobi $β$-Ensemble characteristic polynomial

“…For β = 2 and real τ the random variable X 2 (τ) has a representation in terms of the principal value sum of points of a determinantal point process as proven by Qiu [33] (this is expected to hold for complex τ as well). For general β > 0 we expect an analogous representation in terms of the principal value sum of the eigenvalues of a certain stochastic operator, see [35], [27]. Although, as far as we are aware, this has not been worked out explicitly, it might be possible to obtain using the techniques of [35], [27].…”

“…For general β > 0 we expect an analogous representation in terms of the principal value sum of the eigenvalues of a certain stochastic operator, see [35], [27]. Although, as far as we are aware, this has not been worked out explicitly, it might be possible to obtain using the techniques of [35], [27].…”

“…where the constant is independent of τ. Therefore, (27) equals the following, up to a constant factor independent of τ:…”

Random entire functions from random polynomials with real zeros

“…We note that the papers [43,24,22] also provide quantitative convergence rates to the limiting entire functions studied there. Furthermore, many remarkable properties of these functions have been studied: representations as principal value products [10,9,43], equivalent descriptions in terms of stochastic equations [22], Taylor coefficients given in terms of iterated stochastic integrals [43,24], which are related to some classical identities for Brownian motion [12,37], and explicit moment formulae [43,24]. For generic random Laguerre-P ólya functions that we consider in this paper such precise results are highly unlikely to exist and in principle the most one could hope for is a convergence statement of the sort we prove here.…”

“…Their approach is based on viewing the random characteristic polynomial as a Fredholm determinant of an appropriate stochastic operator. This approach was then also followed in [24] to study two families of random entire functions arising from random matrices which we call here the stochastic Hua-Pickrell (this generalises the stochastic zeta function) and stochastic Bessel functions. Finally, the scaling limit of the characteristic polynomial of Gaussian matrices at the soft edge was studied in [22] and the so-called stochastic Airy function was constructed and studied.…”