The Ginibre Ensemble Conditioned on an Overcrowding Event

TL;DR

This work analyzes the complex Ginibre ensemble conditioned on an overcrowding event where exactly $N_c=\lfloor cN\rfloor$ eigenvalues lie outside a disk of radius $R$ with $R^2>1-c$. It provides an explicit asymptotic formula for the overcrowding probability as a product of the outside-hole probability for size $N_c$ and a partition-function–weighted sum, plus a controllable $O\left(\frac{\log^{3}N}{N}\right)$ error. The conditional eigenvalue distribution is shown to decompose into three regions: an interior region behaving like a scaled Ginibre ensemble of size $N-N_c$, an exterior region matching a full Ginibre ensemble, and a boundary layer that, under $N$-scale rescaling, converges to the determinantal process with kernel $K_{\mathcal X}$ on the right half-plane, mirroring hard-wall Ginibre limits. These results extend hole-event analyses to overcrowding, connect to hard-wall determinantal structures, and offer a detailed description of finite-size conditioning effects in two-dimensional Coulomb gas models.

Abstract

We look at the eigenvalues of the complex Ginibre Ensemble of random matrices consisting of $N$ eigenvalues. We study the event that for $ {c \in [0,1]}$, $\lfloor cN \rfloor$ of the eigenvalues are located outside of a disk of radius $ R \in (\sqrt{1-c},1)$. Except for the case $c=1$ the eigenvalue process conditioned on this event is not determinantal. Nevertheless we are able to obtain asymptotic estimates of the probability of the event, and describe the conditional distribution in three spatial regions. For $ \{ λ\in \mathbb{C} : \big| λ\big| <R\}, \{λ\in \mathbb{C} : \big| λ\big| > R+ε\} $ the conditional distribution is asymptotically that of a Ginibre ensemble. Meanwhile, near the boundary of the disk, after rescaling by a factor of order $ N$, it tends to the determinantal point process that appears in the limit of the Ginibre ensemble near a hard wall in Seo arXiv:2010.08818 [math-ph] .

Figures (1)

• Figure 1: Illustration of the conditional distribution of the eigenvalues of the Ginibre ensemble of size 4000, conditioned on the event that 90% of them satisfy $\left|\lambda\right|\geq R=0.7$, showing the three different regions of the distribution: inside the disk of radius 0.7, on the boundary of the disk and outside the disk.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Proposition 2.1
• Proposition 2.2
• Lemma 2.1
• Remark
• proof : Proof of Lemma \ref{['first bound']}
• Lemma 2.2
• proof