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 $D(s,γ):=\det(I-γK_s\upharpoonright_{L^2(-1,1)})$, $0\leqγ\leq 1$. The operator $K_s$ acts with kernel $K_s(x,y)=\sin(s(x-y))/(π(x-y))$ and $D(s,γ)$ 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