Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Improved discrepancy for the planar Coulomb gas at low temperatures

Felipe Marceca, José Luis Romero

Abstract

We study the planar Coulomb gas in the regime where the inverse temperature $β_n$ grows at least logarithmically with respect to the number of particles $n$ (freezing regime, $β_n\gtrsim \log n$). We show that, almost surely for large $n$, the discrepancy between the number of particles in any microscopic region and their expected value (given with adequate precision by the equilibrium measure) is, up to log factors, of the order of the perimeter of the observation window. The estimates are valid throughout the whole droplet (the region where the particles accumulate), and are particularly interesting near the boundary, while in the bulk they offer technical improvements over known results. Our work builds on recent results on equidistribution at low temperatures and improves on them by providing refined spectral asymptotics for certain Toeplitz operators on the range of the erfc-kernel (sometimes called Faddeeva or plasma dispersion kernel).

Improved discrepancy for the planar Coulomb gas at low temperatures

Abstract

We study the planar Coulomb gas in the regime where the inverse temperature grows at least logarithmically with respect to the number of particles (freezing regime, ). We show that, almost surely for large , the discrepancy between the number of particles in any microscopic region and their expected value (given with adequate precision by the equilibrium measure) is, up to log factors, of the order of the perimeter of the observation window. The estimates are valid throughout the whole droplet (the region where the particles accumulate), and are particularly interesting near the boundary, while in the bulk they offer technical improvements over known results. Our work builds on recent results on equidistribution at low temperatures and improves on them by providing refined spectral asymptotics for certain Toeplitz operators on the range of the erfc-kernel (sometimes called Faddeeva or plasma dispersion kernel).
Paper Structure (21 sections, 22 theorems, 169 equations)

This paper contains 21 sections, 22 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Discrepancies near the boundary
  3. Discrepancies in the bulk
  4. Technical overview and methods
  5. Further remarks
  6. Preliminaries
  7. Notation and basic facts
  8. Scaling limits
  9. Boundary regularity
  10. The spectrum of erfc Toeplitz operators
  11. The Bargmann transform
  12. Comparing erfc and Bargmann concentration operators
  13. Spectral deviation for Bargmann concentration operators
  14. Spectral deviation for erfc concentration operators
  15. Discrepancy at the boundary
  16. ...and 6 more sections

Key Result

Theorem 1.1

Assume that $Q$ satisfies Conditions c1 to c7, $\beta_n\ge c \log n$ with $c>0$, and $\Omega\subseteq \mathbb{C}$ is a compact domain with measure $|\Omega|\ge 2$ and connected boundary. Then there exists a deterministic constant $C=C(c,Q)$ such that almost surely, (Here, although $\Delta Q$ is only defined in a neighborhood $U$ given by Condition c5, for $p\in U^c$ we interpret $\Delta Q(p) |(p

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Microscopic direction and position
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 32 more