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Electrostatic computations for statistical mechanics and random matrix applications

Sung-Soo Byun, Peter J. Forrester

TL;DR

The article surveys electrostatic methods as a unifying framework for problems in statistical mechanics and random matrix theory, highlighting explicit potentials for uniform-background configurations, surface-charge problems, and Green functions. It uses analytic continuation in the Riesz exponent s and conformal-map techniques to derive closed-form potentials for balls, ellipsoids, and other domains, enabling precise leading-order energy and free-energy asymptotics and linking these to random matrix ensembles such as Ginibre and its elliptic/induced variants. The work connects macroscopic electrostatics to microscopic Coulomb-gas descriptions, producing predictions for density profiles, fluctuation formulas, and hole probabilities, with broad applicability to configurations in arbitrary dimensions and to 2D log-gases with background. Overall, it demonstrates how electrostatics yields tractable, exact, or asymptotically exact results that illuminate both statistical-mechanical integrals and spectral properties of non-Hermitian and unitary random matrices.

Abstract

Although for the most part classical, the topic of electrostatics finds to this day new applications. In this review we highlight several theoretical results on electrostatics, chosen to both illustrate general principles, and for their application in statistical mechanics and random matrix settings. The theoretical results include electrostatic potentials and energies associated with balls and hyperellipsoids in general dimension, the use of conformal mappings in two-dimensions, and the balayage measure. A number of explicit examples of their use in predicting the leading asymptotic form of certain configuration integrals and particle density in particular statistical mechanical systems are given, as well as with regards to questions relating to fluctuation formulas and (conditioned) gap probabilities.

Electrostatic computations for statistical mechanics and random matrix applications

TL;DR

The article surveys electrostatic methods as a unifying framework for problems in statistical mechanics and random matrix theory, highlighting explicit potentials for uniform-background configurations, surface-charge problems, and Green functions. It uses analytic continuation in the Riesz exponent s and conformal-map techniques to derive closed-form potentials for balls, ellipsoids, and other domains, enabling precise leading-order energy and free-energy asymptotics and linking these to random matrix ensembles such as Ginibre and its elliptic/induced variants. The work connects macroscopic electrostatics to microscopic Coulomb-gas descriptions, producing predictions for density profiles, fluctuation formulas, and hole probabilities, with broad applicability to configurations in arbitrary dimensions and to 2D log-gases with background. Overall, it demonstrates how electrostatics yields tractable, exact, or asymptotically exact results that illuminate both statistical-mechanical integrals and spectral properties of non-Hermitian and unitary random matrices.

Abstract

Although for the most part classical, the topic of electrostatics finds to this day new applications. In this review we highlight several theoretical results on electrostatics, chosen to both illustrate general principles, and for their application in statistical mechanics and random matrix settings. The theoretical results include electrostatic potentials and energies associated with balls and hyperellipsoids in general dimension, the use of conformal mappings in two-dimensions, and the balayage measure. A number of explicit examples of their use in predicting the leading asymptotic form of certain configuration integrals and particle density in particular statistical mechanical systems are given, as well as with regards to questions relating to fluctuation formulas and (conditioned) gap probabilities.
Paper Structure (27 sections, 8 theorems, 104 equations)

This paper contains 27 sections, 8 theorems, 104 equations.

Table of Contents

  1. Scope of the review
  2. Coulomb potentials and the one-component plasma model
  3. One-component Riesz potential models
  4. Outline of the remainder of the article
  5. Background as an analytic continuation
  6. Riesz gas on a circle
  7. Higher dimensions
  8. Explicit functional forms of electrostatic potentials due to a uniform background
  9. The Coulomb potential with ball geometry
  10. Coulomb potential with hyperellipsoid geometry
  11. Coulomb potential in the case of a rectangular cuboid and rectangle
  12. The domain $\Omega$ as implied by the potential
  13. Applications
  14. Non-intersecting Brownian particles on a line
  15. Complex Ginibre random matrix ensembles
  16. ...and 12 more sections

Key Result

Proposition 3.1

For the ball geometry as specified above, with the ball filled by a uniform background charge density $-\rho_b$, total charge $-N$, the electrostatic potential $V(\vec{r}) = V(r)$ is given by where $\Phi_{\rm d} (r) = \Phi_{\rm d} (\vec{r},\vec{0})$ for any vector $\vec{r}$ such that $\vec{r} = r$.

Theorems & Definitions (15)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Remark 4.1
  • Proposition 4.3
  • ...and 5 more