Free energy expansions of a conditional GinUE and large deviations of the smallest eigenvalue of the LUE

We consider a planar Coulomb gas ensemble of size N with the inverse temperature β=2 and external potential Q(z)=|z|²-2c log|z-a|, where $c>0$ and a ∈ C. Equivalently, this model can be realised as N eigenvalues of the complex Ginibre matrix of size (c+1) N × (c+1) N conditioned to have deterministic eigenvalue a with multiplicity $cN$. Depending on the values of c and a, the droplet reveals a phase transition: it is doubly connected in the post-critical regime and simply connected in the pre-critical regime. In both regimes, we derive precise large-N expansions of the free energy up to the $O(1)$ term, providing a non-radially symmetric example that confirms the Zabrodin-Wiegmann conjecture made for general planar Coulomb gas ensembles. As a consequence, our results provide asymptotic behaviours of moments of the characteristic polynomial of the complex Ginibre matrix, where the powers are of order $O(N)$. Furthermore, by combining with a duality formula, we obtain precise large deviation probabilities of the smallest eigenvalue of the Laguerre unitary ensemble. Our proof is based on a refined Riemann-Hilbert analysis for planar orthogonal polynomials using the partial Schlesinger transform.