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On fluctuations of Coulomb systems and universality of the Heine distribution

Yacin Ameur, Joakim Cronvall

TL;DR

This work analyzes two universality classes in planar Coulomb gas ensembles at $eta=2$ under smooth outpost and spectral-gap potentials. It proves that fluctuations near an outpost follow a Heine distribution with parameters determined by the conformal-capacity data and Dirichlet problem, while fluctuations across a spectral gap decompose into the difference of two Heine variables with $n$-dependent parameters, plus an independent Gaussian component for linear statistics. A key technical achievement is a new asymptotic formula for monic orthogonal polynomials in the bifurcation regime, together with a variant of the limit Ward identities, enabling precise control of fluctuations via decomposition into $oldsymbol{ extG}$ and $oldsymbol{ extH}$ components. The results illuminate universal fluctuation behavior in 2D Coulomb systems with nontrivial topology (outposts and gaps) and connect to Polyakov-Alvarez-type large-$n$ expansions of the free energy, with implications for spectral geometry and random normal matrices.

Abstract

We consider a class of external potentials on the complex plane $\mathbb{C}$ for which the coincidence set to the obstacle problem contains a Jordan curve in the exterior of the droplet. We refer to this curve as a spectral outpost. We study the corresponding Coulomb gas at $β=2$. Generalizing recent work in the radially symmetric case, we prove that the number of particles which fall near the spectral outpost has an asymptotic Heine distribution, as the number of particles $n\to\infty$.We also consider a class of potentials with disconnected droplets whose connected components are separated by a ring-shaped spectral gap. We prove that the fluctuations of the number of particles that fall near a given component has an asymptotic discrete normal distribution, which depends on $n$. For the case of disconnected droplets we also consider fluctuations of general smooth linear statistics and show that they tend to distribute as the sum of a Gaussian field and an independent, oscillatory, discrete Gaussian field. Our techniques involve a new asymptotic formula on the norm of monic orthogonal polynomials in the bifurcation regime and a variant of the method of limit Ward identities of Ameur, Hedenmalm, and Makarov.

On fluctuations of Coulomb systems and universality of the Heine distribution

TL;DR

This work analyzes two universality classes in planar Coulomb gas ensembles at under smooth outpost and spectral-gap potentials. It proves that fluctuations near an outpost follow a Heine distribution with parameters determined by the conformal-capacity data and Dirichlet problem, while fluctuations across a spectral gap decompose into the difference of two Heine variables with -dependent parameters, plus an independent Gaussian component for linear statistics. A key technical achievement is a new asymptotic formula for monic orthogonal polynomials in the bifurcation regime, together with a variant of the limit Ward identities, enabling precise control of fluctuations via decomposition into and components. The results illuminate universal fluctuation behavior in 2D Coulomb systems with nontrivial topology (outposts and gaps) and connect to Polyakov-Alvarez-type large- expansions of the free energy, with implications for spectral geometry and random normal matrices.

Abstract

We consider a class of external potentials on the complex plane for which the coincidence set to the obstacle problem contains a Jordan curve in the exterior of the droplet. We refer to this curve as a spectral outpost. We study the corresponding Coulomb gas at . Generalizing recent work in the radially symmetric case, we prove that the number of particles which fall near the spectral outpost has an asymptotic Heine distribution, as the number of particles .We also consider a class of potentials with disconnected droplets whose connected components are separated by a ring-shaped spectral gap. We prove that the fluctuations of the number of particles that fall near a given component has an asymptotic discrete normal distribution, which depends on . For the case of disconnected droplets we also consider fluctuations of general smooth linear statistics and show that they tend to distribute as the sum of a Gaussian field and an independent, oscillatory, discrete Gaussian field. Our techniques involve a new asymptotic formula on the norm of monic orthogonal polynomials in the bifurcation regime and a variant of the method of limit Ward identities of Ameur, Hedenmalm, and Makarov.

Paper Structure

This paper contains 34 sections, 20 theorems, 215 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Coulomb gas ensembles
  3. Linear statistics
  4. Class of outpost potentials
  5. Examples of outpost potentials
  6. Number of particles near the outpost
  7. Class of spectral gap potentials
  8. Examples of spectral gap potentials
  9. Number of particles near the ring $S\setminus S_{\tau_*}$
  10. A convergence result for smooth linear statistics
  11. Orthogonal polynomials in the bifurcation regime
  12. A surmise on the large-$n$ expansion of the free energy
  13. Comments.
  14. Plan of this paper
  15. An approximation formula for monic orthogonal polynomials
  16. ...and 19 more sections

Key Result

Lemma 1.1

Suppose that $X\sim \operatorname{He}(\theta,q)$. The cumulant generating function $F_X(s)=\log{\mathbb E}\exp(sX)$ is given by

Figures (2)

  • Figure 1: Coincidence set $S^*$ for an outpost potential
  • Figure 2: Droplet of a spectral gap potential

Theorems & Definitions (37)

  • Remark 1
  • Remark 2
  • Lemma 1.1
  • Theorem 1.2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 1.3
  • Remark 7
  • ...and 27 more