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

Abstract

We consider a planar Coulomb gas ensemble of size with the inverse temperature and external potential , where and . Equivalently, this model can be realised as eigenvalues of the complex Ginibre matrix of size conditioned to have deterministic eigenvalue with multiplicity . Depending on the values of and , 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‐ expansions of the free energy up to the 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 . 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.