  In this paper we continue our analysis \cite{BDIK} of the determinant
$\det(I-\gamma K_s),\gamma\in(0,1)$ where $K_s$ is the trace class operator
acting in $L^2(-1,1)$ with kernel $K_s(\lambda,\mu)=\frac{\sin
s(\lambda-\mu)}{\pi(\lambda-\mu)}$. In \cite{BDIK} various key asymptotic
results were stated and utilized, but without proof: Here we provide the proofs
(see Theorem 1.2 and Proposition 1.3 below).
