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Partition functions of two-dimensional Coulomb gases with circular root- and jump-type singularities

Kohei Noda

Abstract

In this paper, we study the random polynomial $p_n(ρ):=\prod_{j=1}^n (|z_j|-ρ)$, where the points $\{z_j\}_{j=1}^n$ are the eigenvalue moduli of random normal matrices with a radially symmetric potential. We establish precise large $n$ asymptotic expansions for the moment generating function \[ \mathbb{E}\!\left[e^{\tfrac{u}π\mathrm{Im}\log p_n(ρ)}\, e^{a\,\mathrm{Re}\log p_n(ρ)}\right], \qquad u\in\mathbb{R}, \; a>-1, \] where $ρ>0$ lies in the bulk of the spectral droplet. The asymptotic expansion is expressed in terms of parabolic cylinder functions, which confirms a conjecture of Byun and Charlier. This also provides the first free energy expansion of two-dimensional Coulomb gases with general circular root- and jump-type singularities. While the $a=0$ case has already been widely studied in the literature due to its relation to counting statistics, we also obtain new results for this special case.

Partition functions of two-dimensional Coulomb gases with circular root- and jump-type singularities

Abstract

In this paper, we study the random polynomial , where the points are the eigenvalue moduli of random normal matrices with a radially symmetric potential. We establish precise large asymptotic expansions for the moment generating function \[ \mathbb{E}\!\left[e^{\tfrac{u}π\mathrm{Im}\log p_n(ρ)}\, e^{a\,\mathrm{Re}\log p_n(ρ)}\right], \qquad u\in\mathbb{R}, \; a>-1, \] where lies in the bulk of the spectral droplet. The asymptotic expansion is expressed in terms of parabolic cylinder functions, which confirms a conjecture of Byun and Charlier. This also provides the first free energy expansion of two-dimensional Coulomb gases with general circular root- and jump-type singularities. While the case has already been widely studied in the literature due to its relation to counting statistics, we also obtain new results for this special case.

Paper Structure

This paper contains 8 sections, 12 theorems, 141 equations, 1 figure.

Table of Contents

  1. Introduction and statement of results
  2. Global analysis part for the proof of Theorem \ref{['theorem:calEn expansion']}
  3. Local asymptotic analysis for the proof of Theorem \ref{['theorem:calEn expansion']}
  4. Asymptotic expansion of S_2
  5. Proof of Theorem \ref{['theorem:calEn expansion']}
  6. Consistency between Theorem \ref{['theorem:asymptotic expansion of counting statistics']} and C2021 FH
  7. Consistency between Theorem \ref{['theorem:calEn expansion']} and BC2022
  8. Euler-Maclaurin formula

Key Result

Theorem 1.2

Let $\rho\in(0,r_1)$, $\alpha>-1$, $u\in\mathds{R}$, and $a=0$. Under Assumptions Assumption_Q, there exists $\delta>0$ such that, as $n\to+\infty$, we have uniformly for $u\in\{z\in\mathds{C}:|z-x|\leq\delta\}$, where

Figures (1)

  • Figure 1: The plot (a) shows $a\mapsto C_3$ (black line) and its comparison $a\mapsto \log\mathcal{E}_{n,u,a}-(C_1\,n+C_2\,\sqrt{n})$, where $Q(z)=0.2|z|^2+0.2345|z|^3, \alpha=0.667, u = 1.56$, $\rho=0.71r_1$, $n=10$ (red, dotted line), $n = 40$ (blue, dot-dashed line), and $n = 160$ (purple, dashed line) The plot (b) shows $\rho\mapsto C_3$ (black line) and its comparison $\rho\mapsto \log\mathcal{E}_{n,u,a}-(C_1\,n+C_2\,\sqrt{n})$, where $Q,\alpha,u$ are same as before, $a=1.25$, and $n=100$ (red, dotted line), $n = 300$ (blue, dot-dashed line), and $n = 600$ (purple, dashed line).

Theorems & Definitions (22)

  • Remark 1.1
  • Theorem 1.2: Counting statistics
  • Remark 1.3: Consistency with Theorem 1.1 in C2021 FH
  • Corollary 1.4
  • Theorem 1.5: Counting statistics and root-type statistics
  • Remark 1.6
  • Theorem 1.7: Partition function with circular- and root-type singularities
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 12 more