The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem

Blackstone, Elliot
Charlier, Christophe
Lenells, Jonatan

Publication date

January 2021

Abstract

The Bessel process models the local eigenvalue statistics near $0$ of certain large positive definite matrices. In this work, we consider the probability \begin{align*} \mathbb{P}\Big( \mbox{there are no points in the Bessel process on } (0,x_{1})\cup(x_{2},x_{3})\cup\cdots\cup(x_{2g},x_{2g+1}) \Big), \end{align*} where $0<x_{1}<\cdots<x_{2g+1}$ and $g \geq 0$ is any non-negative integer. We obtain asymptotics for this probability as the size of the intervals becomes large, up to and including the oscillations of order $1$. In these asymptotics, the most intricate term is a one-dimensional integral along a linear flow on a $g$-dimensional torus, whose integrand involves ratios of Riemann $\theta$-functions associated to a genus $g$ Riemann ...

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The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem

Abstract

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The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem

Abstract

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