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

TL;DR

This work analyzes a two-dimensional Coulomb gas at inverse temperature $\beta=2$ with a log-singular external potential $Q(z)=|z|^2-2c\log|z-a|$, realized as a conditioned complex Ginibre ensemble. It establishes a unified, dual perspective linking GinUE moments, the induced GinUE partition function $Z_N(a,c)$, and Laguerre unitary ensemble smallest-eigenvalue statistics, enabling precise large-$N$ expansions of the free energy up to $O(1)$ terms. The authors derive explicit post- and pre-critical free-energy formulas, identify the Euler characteristic of the droplets, and reveal a critical regime governed by Tracy–Widom fluctuations, connected via a duality to LUE large deviations. The results confirm Zabrodin–Wiegmann predictions for a non-radially symmetric example, derive asymptotics for moments of GinUE characteristic polynomials, and yield sharp large-deviation probabilities for the LUE smallest eigenvalue through a refined two-dimensional RH analysis with a partial Schlesinger transform.

Abstract

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 $a \in \mathbb{C}$. 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 $O(1)$ 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.

Figures (8)

• Figure 1: Illustration of the droplet, where $c=9/16$. The black dot indicates the point $a.$
• Figure 2: Illustration of the Marchenko-Pastur law \ref{['MP']} and constrained spectral density \ref{['constrained LUE density']}, where $\alpha=5$. Here, the vertical (full red) line indicates the hard wall $x=t$ of the LUE.
• Figure 3: The plot shows the graph of the energy $a \mapsto I_Q[\sigma_Q]$, where $c=9/16$. Here, the dotted vertical line represents $a=a_{\rm cri}=1/2$. The graph (full line) for $a<a_{\rm cri}$ follows \ref{['energy post']}, while for $a>a_{\rm cri}$ it follows \ref{['energy pre']}. The dotted line for $a>a_{\rm cri}$ is the continuation of the graph \ref{['energy post']}.
• Figure 4: The full red line represents the graph of the rate function $t \mapsto \Phi(t;\alpha)$, where $\alpha=16/9$ and $t \geq \lambda_-=4/9$. The blue dots indicate the values of the function $t \mapsto S(t)-S(\lambda_-)$. Here, one can also observe the behaviours \ref{['rate function 0']} and \ref{['rate function infty']} in the left and right figures, respectively.
• Figure 5: The plot shows deformations of the droplet. The leftmost $(a=0)$ and rightmost $(a=\infty)$ are the rotationally symmetric cases, for which we use the associated partition functions as reference. The thick lines indicate the integral domains in \ref{['ZN ref post']} and \ref{['ZN ref pre']}, respectively.

Theorems & Definitions (59)

• Proposition 1.1: Duality relation
• Remark 1.2: Potential with a hard edge
• Definition 1.3: Critical scaling regime
• Proposition 2.1: Evaluation of the energy
• Theorem 2.2: Free energy expansion for the post- and pre-critical cases
• Remark 2.3: Regularized determinant of Laplacian
• Remark 2.4: Absence of the $O(\sqrt{N})$ term for $\beta=2$
• Proposition 2.5: Free energy expansion for the critical regime
• Remark 2.6: Free energy expansion under the topology transitions
• Theorem 2.7: Moments of characteristic polynomial of the GinUE