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On the Maximum of the Potential of a General Two-Dimensional Coulomb Gas

Luke Peilen

Abstract

We determine the leading order of the maximum of the random potential associated to a two-dimensional Coulomb gas for general $β$ and general confinement potential, extending the recent result of Lambert-Leblé-Zeitouni. In the case $β=2$, this corresponds to the (centered) log-characteristic polynomial of either the Ginibre random matrix ensemble for $V(x)=\frac{|x|^2}{2}$ or a more general normal matrix ensemble. The result on the leading order asymptotics for the maximum of the log-characteristic polynomial is new for random normal matrices. We rely on connections with the classical obstacle problem and the theory of Gaussian Multiplicative Chaos. We make use of a new concentration result for fluctuations of $C^{1,1}$ linear statistics which may be of independent interest.

On the Maximum of the Potential of a General Two-Dimensional Coulomb Gas

Abstract

We determine the leading order of the maximum of the random potential associated to a two-dimensional Coulomb gas for general and general confinement potential, extending the recent result of Lambert-Leblé-Zeitouni. In the case , this corresponds to the (centered) log-characteristic polynomial of either the Ginibre random matrix ensemble for or a more general normal matrix ensemble. The result on the leading order asymptotics for the maximum of the log-characteristic polynomial is new for random normal matrices. We rely on connections with the classical obstacle problem and the theory of Gaussian Multiplicative Chaos. We make use of a new concentration result for fluctuations of linear statistics which may be of independent interest.
Paper Structure (14 sections, 10 theorems, 76 equations)

This paper contains 14 sections, 10 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. The Model
  3. Main Result
  4. Assumptions
  5. Connection with Literature
  6. Proof Structure and Outline of Paper
  7. Acknowledgements
  8. Proof of the Main Theorem
  9. Infrastructure from the Quadratic Case
  10. LLZ23, Preliminaries on Coulomb Gases
  11. LLZ23, Upper bound for Law of Large Numbers
  12. LLZ23, Lower bound for Law of Large Numbers and Gaussian Multiplicative Chaos
  13. Proof of Main Theorem
  14. Main Technical Estimate

Key Result

Theorem 1

Let $V$ satisfy assumptions (A1)-(A3), given in $\S Assumptions$. Suppose additionally that $\Sigma$ is one-cut and that $V \in C^5(\mathbb{R}^2)$. Let $r>0$ be such that $\mathsf{D}(x,r) \subseteq \Sigma$ and $\mathsf{D}(x,r) \cap \partial \Sigma=\emptyset$. Then, we have as $N \rightarrow +\infty$, with the convergence in probability.

Theorems & Definitions (16)

  • Theorem 1
  • Proposition 1
  • Proposition 1.1: C98; Theorem 2; CK80
  • proof : Proof of Theorem \ref{['theo']}
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4: Control of $T_0$
  • ...and 6 more