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A maximum principle for the Coulomb gas: microscopic density bounds, confinement estimates, and high temperature limits

Eric Thoma

Abstract

We introduce and prove a maximum principle for a natural quantity related to the $k$-point correlation function of the classical one-component Coulomb gas. As an application, we show that the gas is confined to the droplet by a well-known effective potential in dimensions two and higher. We also prove new upper bounds for the particle density in the droplet that apply at any temperature. In particular, we give the first controls on the microscopic point process for high temperature Coulomb gases beyond the mean-field regime, proving that their laws are uniformly tight in the particle number $N$ for any inverse temperatures $β_N$. Furthermore, we prove that limit points are homogeneous mixed Poisson point processes if $β_N\to 0$.

A maximum principle for the Coulomb gas: microscopic density bounds, confinement estimates, and high temperature limits

Abstract

We introduce and prove a maximum principle for a natural quantity related to the -point correlation function of the classical one-component Coulomb gas. As an application, we show that the gas is confined to the droplet by a well-known effective potential in dimensions two and higher. We also prove new upper bounds for the particle density in the droplet that apply at any temperature. In particular, we give the first controls on the microscopic point process for high temperature Coulomb gases beyond the mean-field regime, proving that their laws are uniformly tight in the particle number for any inverse temperatures . Furthermore, we prove that limit points are homogeneous mixed Poisson point processes if .
Paper Structure (14 sections, 24 theorems, 174 equations)

This paper contains 14 sections, 24 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Definitions and assumptions
  3. Main results and comparisons to literature
  4. Paper organization and future directions
  5. A maximum principle for the Coulomb gas
  6. Microscopic point process
  7. Density upper bounds
  8. High temperature limit points
  9. Confinement of the Coulomb gas
  10. Proof of the main result
  11. The distance to vacuum
  12. The thermal equilibrium measure
  13. Proof of Proposition \ref{['p.farfield.simple']} in dimension two
  14. Proof of Proposition \ref{['p.farfield.simple']} in dimension three and higher

Key Result

Theorem 1

One has in an integral sense. In particular, the quantity $e^{\beta \zeta} \rho_1$ is subharmonic on the complement of $\Sigma = \mathrm{supp } \ \mu_\infty$.

Theorems & Definitions (54)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 4
  • Definition 2.1
  • Proposition 2.2
  • ...and 44 more