## The sine process under the influence of a varying potential.(English)Zbl 1404.60010

Summary: We review the authors’ recent work 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 of a Coulomb log-gas with varying external potential or in the bulk scaling analysis of the thinned Gaussian unitary ensemble.{