The Fourier-Bessel Series and Hard Edge Limits

Abstract

The universality classes defined by the Airy and Bessel kernels are two of the most fundamental in random matrices and growth models more generally. Broadly speaking, one often encounters the Airy kernel when studying models where the relevant eigenvalues or particles are unbounded, and the Bessel kernel when examining their constrained counterparts. In this thesis, we analyze two recent problems where the relevant expressions involve a variant of the Airy functions known as the Fourier-Airy series. In both cases, we find that the constrained versions have natural analogues expressible in terms of the Fourier-Bessel series echoing the relationship between the Airy and Bessel kernels. In the first part, we study the hard edge limit of a multilevel extension of the Laguerre β-ensemble at zero temperature. In particular, we show that asymptotically the ensemble is given by Gaussians with covariance matrix expressible in terms of the Fourier-Bessel series. These Gaussians also have an explicit representation as the partition functions of additive polymers arising from a random walk on roots of the Bessel functions. Our approach builds off of techniques introduced by Gorin and Kleptsyn [1] and is rooted in using the theory of dual and associated polynomials to diagonalize transition matrices relating levels of the ensemble. Like the corresponding soft edge limit in the Hermite case studied by Gorin and Kleptsyn, the object we introduce should represent a new universality class for zero temperature random matrices. In the second part, we introduce a new diffusion process which arises as the n → ∞ limit of a Bessel process of dimension d ≥ 2 conditioned upon remaining bounded below one until time n. In addition to being interesting in its own right, we argue that the resulting diffusion process is a natural hard edge counterpart to the Ferrari-Spohn diffusion of [2].