The sine process under the influence of a varying potential

Abstract

We review the authors' recent work \cite{BDIK1,BDIK2,BDIK3} where we obtain the uniform large $s$ asymptotics for the Fredholm determinant $D(s,\gamma ):=\det (I-\gamma K_s{\upharpoonright }_{L^2(-1,1)})$, $0\le \gamma \le 1$. 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}.

7 pages, 1 figure, a contribution to JMP's volume devoted to the memory of LD Faddeev