The sine process under the influence of a varying potential

We review the authors’ recent work \cite{BDIK1,BDIK2,BDIK3} where we obtain the uniform large $s$ asymptotics for the Fredholm determinant $D(s,\gamma ):=\det (I-\gamma K_s{\upharpoonright }_{L^2(-1,1)})$, $0\leq\gamma\leq 1$\mathrm{.}Theoperator$K_s$actswithkernel$K_s(x,y)=\sin(s(x-y))/(\pi(x-y))$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}.