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Law of large numbers for the maximum of the two-dimensional Coulomb gas potential

Gaultier Lambert, Thomas Leblé, Ofer Zeitouni

Abstract

We derive the leading order asymptotics of the logarithmic potential of a two dimensional Coulomb gas at arbitrary positive temperature. The proof is based on precise evaluation of exponential moments, and the theory of Gaussian multiplicative chaos.

Law of large numbers for the maximum of the two-dimensional Coulomb gas potential

Abstract

We derive the leading order asymptotics of the logarithmic potential of a two dimensional Coulomb gas at arbitrary positive temperature. The proof is based on precise evaluation of exponential moments, and the theory of Gaussian multiplicative chaos.
Paper Structure (34 sections, 28 theorems, 155 equations)

This paper contains 34 sections, 28 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Setting and main result
  3. Connections with the literature
  4. Comments on the strategy.
  5. Sketch of the proof
  6. Open problems
  7. Notation
  8. Preliminaries
  9. Re-writing the energy and the Gibbs measure
  10. Energy at global and local scales
  11. Fluctuations
  12. Re-writing the Laplace transform
  13. Comparison of partition functions
  14. Law of large numbers: upper bound
  15. An auxiliary linear statistics
  16. ...and 19 more sections

Key Result

Theorem 1

For all $r \in (0,1)$ we have:

Theorems & Definitions (54)

  • Theorem 1: LLN for the max of the 2D Coulomb gas potential
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3: Local laws for $\mathsf{EnerPts}$
  • Lemma 2.4
  • proof
  • Lemma 2.5: Laplace transforms as ratio of partition functions
  • proof
  • Proposition 2.6: Main comparison result
  • Remark 2.7
  • ...and 44 more