The sine process under the influence of a varying potential

We review the authors' recent work where we obtain the uniform large s asymptotics for the Fredholm determinant D (s ,γ ) ≔det(I -γ K_s↾_{L²(-1,1 )} ) , 0 ≤ γ ≤ 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 of a Coulomb log-gas with varying external potential or in the bulk scaling analysis of the thinned Gaussian unitary ensemble.