# Large gap asymptotics for Airy kernel determinants with discontinuities

## TL;DR

This paper derives large gap asymptotics for Airy kernel determinants with multiple discontinuities, revealing their asymptotic factorization and explicit constants related to eigenvalue distributions near soft edges.

## Contribution

It provides the first comprehensive asymptotic analysis for multi-point Airy kernel determinants with discontinuities, extending previous single-point results.

## Key findings

- Asymptotic expression as product of single-point determinants
- Explicit constant pre-factor related to covariance of counting functions
- Results applicable to eigenvalue distributions in random matrix theory

## Abstract

We obtain large gap asymptotics for Airy kernel Fredholm determinants with any number $m$ of discontinuities. These $m$-point determinants are generating functions for the Airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles. Our main result is that the $m$-point determinants can be expressed asymptotically as the product of $m$ $1$-point determinants, multiplied by an explicit constant pre-factor which can be interpreted in terms of the covariance of the counting function of the process.