Transition Analysis: From the Airy and Pearcey Kernels to the Sine Kernel

TL;DR

This paper analyzes the transition between Airy, Pearcey, and sine kernels in Random Matrix Theory, providing detailed asymptotic expansions that describe how these kernels evolve under rescaling at different spectral points.

Contribution

It offers a comprehensive asymptotic analysis of the transitions between the three universal kernels, elucidating the fluctuation behavior in the process.

Findings

Asymptotic expansions for Airy and Pearcey kernels approaching the sine kernel.

Explicit descriptions of fluctuations during kernel transitions.

Complete characterization of kernel behavior at spectrum edges, cusps, and bulk points.

Abstract

We study transitions between the three universal limiting kernels Airy, Pearcey and sine kernel, arising in Random Matrix Theory at edge, cusp and bulk points of the spectrum. Under appropriate rescalings, we provide complete asymptotic expansions of the extended Airy kernel and the extended Pearcey kernel approaching the extended sine kernel, expliciting the fluctuations.