Abstract
We consider a planar Coulomb gas ensemble of size N with the inverse temperature beta = 2 and external potential Q(z) = |z|(2)-2c log |z-a|, where c > 0 and a E C. Equivalently, this model can be realised as N eigenvalues of the complex Ginibre matrix of size (c + 1)N x (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 behaviour 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. A key ingredient for the proof lies in the fine asymptotic behaviour of a planar orthogonal polynomial, extending a result of Betola et al. This result holds its own interest and is based on a refined Riemann-Hilbert analysis using the partial Schlesinger transform.

https://doi.org/10.1002/cpa.70005