Asymptotic expansion of the hard-to-soft edge transition

TL;DR

This paper derives a detailed asymptotic expansion for the transition from hard to soft edge in random matrix theory, confirming a recent conjecture and advancing understanding of spectral edge behaviors.

Contribution

It provides the first full asymptotic expansion of the symmetrically transformed Bessel kernel at the hard-to-soft edge transition, confirming a conjecture by Bornemann.

Findings

Established a full asymptotic expansion for the Bessel kernel

Resolved a conjecture on the hard-to-soft edge transition

Enhanced understanding of spectral edge asymptotics

Abstract

By showing that the symmetrically transformed Bessel kernel admits a full asymptotic expansion for large parameter, we establish a hard-to-soft edge transition expansion. This resolves a conjecture recently proposed by Bornemann.