Confluent hypergeometric kernel determinant on multiple large intervals

TL;DR

This paper derives large gap asymptotics for the confluent hypergeometric point process over multiple intervals, revealing oscillatory behavior and involving theta functions, with improved estimates under certain Diophantine conditions.

Contribution

It provides a comprehensive asymptotic analysis of the confluent hypergeometric kernel determinant on multiple large intervals, including oscillatory terms and error estimates.

Findings

Asymptotic formula involving theta functions for large gaps

Improved error estimates under Diophantine and ergodic conditions

Precise large gap asymptotics for the case n=1

Abstract

The confluent hypergeometric point process represents a universality class which arises in a variety of different but related areas. It particularly describes the local statistics of eigenvalues in the bulk of spectrum near a Fisher-Hartwig singular point for a broad class of unitary ensembles. It is the aim of this work to investigate large gap asymptotics of this process over a union of disjoint intervals $\cup_{j=0}^{n}(sa_j,sb_j)$, where $a_0<b_0<\dots<a_m<0<b_m<\dots<a_n<b_n$ for some $0\leq m \leq n$. As $s\to +\infty$, we establish a general asymptotic formula up to and including the oscillatory term of order $1$, which involves a $\theta$-functions-combination integral along a linear flow on an $n$-dimensional torus. If the linear flow has ``good Diophantine properties'' or the ergodic properties, we further improve the error estimate or the leading term for the asymptotics of the integral. These results can be combined for the case $n=1$, which lead to a precise large gap asymptotics up to an undetermined constant.