Some formulas of relative asymptotics for the partition function in the multi-cut case

In the case of a regular polynomial potential $Q(x)$ for its small perturbation of type $Q(x;t)=Q(x)+tx$, $t$ – a small parameter, we obtain some formulas of "relative asymptotics" for the corresponding partition function $Z_n(t)$ as $n\to\infty$ (see attached file).