The sine process under the influence of a varying potential

TL;DR

This paper reviews recent work on the asymptotic behavior of a Fredholm determinant related to the sine process under a varying potential, with implications for random matrix theory and Coulomb gases.

Contribution

It provides uniform large s asymptotics for the Fredholm determinant associated with the sine kernel under a varying potential, extending previous results.

Findings

Derived uniform large s asymptotics for D(s,γ)

Connected results to Dyson's Coulomb log-gas model

Analyzed the impact of a varying external potential on the sine process

Abstract

We review the authors' recent work \cite{BDIK1,BDIK2,BDIK3} where we obtain the uniform large $s$ asymptotics for the Fredholm determinant $D(s,\gamma):=\det(I-\gamma K_s\upharpoonright_{L^2(-1,1)})$, $0\leq\gamma\leq 1$. The operator $K_s$ acts with kernel $K_s(x,y)=\sin(s(x-y))/(\pi(x-y))$ and $D(s,\gamma)$ appears for instance in Dyson's model \cite{Dyson2} of a Coulomb log-gas with varying external potential or in the bulk scaling analysis of the thinned GUE \cite{BP}.