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Birth of a gap: Critical phenomena in 2D Coulomb gas

Matthias Allard, Sampad Lahiry

TL;DR

This work analyzes a 2D Coulomb gas at $\beta=2$ with a radially symmetric confining potential that induces a gap on a circle inside the droplet. It develops a sharp large-$n$ expansion of the partition function, unveiling a novel universal $n^{1/4}$ term accompanied by a Pearcey-like contribution, and establishes a double-scaling limit for the local correlation kernel near the gap, yielding a new universal kernel $K_*(\xi_1,\xi_2)$ distinct from the Ginibre kernel. The results bridge macroscopic energy–entropy descriptions with microscopic point-process behavior, revealing a universal structure governed by a Pearcey-type integral and providing precise mean-level spacing scaling $s_n \sim c (\gamma_Q n)^{-1/4}$. The findings suggest broad universality of the critical kernel under radial symmetry and point to rich directions for higher-order criticalities, non-radial potentials, and fluctuations of linear statistics. Overall, the paper advances understanding of critical phenomena in 2D Coulomb gases and introduces a new universal local limit in the bulk near a gap circle.

Abstract

We investigate a family of radially symmetric Coulomb gas systems at inverse temperature $β= 2$. The family is characterised by the property that the density of the equilibrium measure vanishes on a ring at radius $r_*$, which lies strictly inside the droplet. The large $n$ expansion of the logarithm of the partition function is obtained up to a novel $n^{1/4}$ term. We perform a double scaling limit of the correlation kernel at the $n^{1/4}$ scale and obtain a new limiting kernel in the bulk, which differs from the well-known Ginibre kernel.

Birth of a gap: Critical phenomena in 2D Coulomb gas

TL;DR

This work analyzes a 2D Coulomb gas at with a radially symmetric confining potential that induces a gap on a circle inside the droplet. It develops a sharp large- expansion of the partition function, unveiling a novel universal term accompanied by a Pearcey-like contribution, and establishes a double-scaling limit for the local correlation kernel near the gap, yielding a new universal kernel distinct from the Ginibre kernel. The results bridge macroscopic energy–entropy descriptions with microscopic point-process behavior, revealing a universal structure governed by a Pearcey-type integral and providing precise mean-level spacing scaling . The findings suggest broad universality of the critical kernel under radial symmetry and point to rich directions for higher-order criticalities, non-radial potentials, and fluctuations of linear statistics. Overall, the paper advances understanding of critical phenomena in 2D Coulomb gases and introduces a new universal local limit in the bulk near a gap circle.

Abstract

We investigate a family of radially symmetric Coulomb gas systems at inverse temperature . The family is characterised by the property that the density of the equilibrium measure vanishes on a ring at radius , which lies strictly inside the droplet. The large expansion of the logarithm of the partition function is obtained up to a novel term. We perform a double scaling limit of the correlation kernel at the scale and obtain a new limiting kernel in the bulk, which differs from the well-known Ginibre kernel.

Paper Structure

This paper contains 18 sections, 14 theorems, 145 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Presentation of the model
  4. Statement of results
  5. Example of potential
  6. Preliminaries and notations
  7. Asymptotic expansion of the norms $\norm{p_{j,n}}$
  8. Annulus case
  9. Disk case
  10. Asymptotic expansion of $Z_n$
  11. Annulus case
  12. Disk case
  13. Local correlations
  14. Local Kernel estimate
  15. Proof of \ref{['theo2']}
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let the potential $Q$ be such as in pres of model. The large $n$ expansion of the partition function $Z_n$eq:betapart is given as where $I_Q[\mu_Q]$, $E_Q[\mu_Q]$, $\varkappa_Q$ and $\mathcal{P}$ are given, respectively, in eq: energy, eq:entropy, eq:vcon and eq: pearcey.

Figures (5)

  • Figure 1: (Left)$r_0>0$: The droplet $\mathbb{S}$ (blue) is an annulus $\mathbb{A}(r_0,r_1)$ and $\mathbb{T}_{r_*}$, the circle of radius $r_*$ (red), lies inside the annulus. (Right)$r_0=0$: The droplet $\mathbb{S}$ (blue) is a disk of radius $r_1$ and $\mathbb{T}_{r_*}$ (red), lies inside the annulus.
  • Figure 2: Plot of the function $\xi\mapsto\abs{K_*(\xi,0)}$, as defined in equation \ref{['eq: K_*']} for $-6\leqslant \Re(\xi)\leqslant6$, $-8\leqslant \Im(\xi)\leqslant8$. The quantity $\abs{K_*(\xi,0)}^2$ is a measure of the covariance between a point at $z=0$ and a point at $z=\xi$ (in the local referential centred at $r_*$). See \ref{['rem: interpretation']} for some further comment.
  • Figure 3: Plots of the functions $\xi \mapsto \rho(\xi)$, as defined in equation \ref{['eq: def rho']}, and $\xi \mapsto \xi^2/2 + c_1$, where $c_1 = \rho(0)$, shown for comparison over the real interval $\xi \in [-6, 6]$. (left) normal scale. (right)$y$-axis in $\log$ scale.
  • Figure 4: Plots of the functions $\xi \mapsto\rho^{(m)}(\xi)$, defined in \ref{['eq: def rho m']}, and $\xi\mapsto \xi^{2m}/(2m)!+c_m$, with $c_m=\rho^{(m)}(0)$, shown for comparison over the real interval $\xi \in [-6, 6]$. For $m=1,2,3$. $y$-axis in $\log$ scale.
  • Figure 5: (Left)$t=t_{\rm min}$: The droplet (blue) is a disk of radius $R_3$ with the exclusion of a circle of radius $r_*$. (Right)$t>t_{\rm min}$: The droplet (blue) is composed of a disk of radius $\mathbb{D}(R_3)=\{0\leqslant \abs{z}\leqslant R_3 \}$ and an annulus $\mathbb{A}(R_2,R_3)=\{R_2\leqslant \abs{z}\leqslant R_3 \}$. The Laplacian of the potential, $\Delta Q_t$, is strictly negative inside the annulus $\mathbb{A}(r_-,r_+)=\{r_-\leqslant \abs{z}\leqslant r_+ \}$ (orange) and vanishes on the boundaries.

Theorems & Definitions (31)

  • Remark 1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2
  • Remark 3
  • Corollary 1
  • Remark 4
  • Definition 1
  • Corollary 2
  • Remark 5
  • ...and 21 more