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 $N$ with the inverse temperature $\beta =2$ and external potential $Q\left(z\right)={\left|z\right|}^2-2c\log |z-a|$, where $c>0$ and $a\in C$. Equivalently, this model can be realised as $N$ eigenvalues of the complex Ginibre matrix of size $(c+1)N\times (c+1)N$ 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 $O\left(1\right)$ 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 $O\left(N\right)$. 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.