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