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Free energy of spherical Coulomb gases with point charges

Sung-Soo Byun, Nam-Gyu Kang, Seong-Mi Seo, Meng Yang

Abstract

We consider two-dimensional Coulomb gases on the Riemann sphere with determinantal or Pfaffian structures, under external potentials that are invariant under rotations around the axis connecting the north and south poles, and with microscopic point charges inserted at the poles. These models can be interpreted as Coulomb gases on the complex plane with weakly confining potentials, where the associated droplet is the entire complex plane. For these models, we derive precise asymptotic expansions of the free energies, including the constant terms.

Free energy of spherical Coulomb gases with point charges

Abstract

We consider two-dimensional Coulomb gases on the Riemann sphere with determinantal or Pfaffian structures, under external potentials that are invariant under rotations around the axis connecting the north and south poles, and with microscopic point charges inserted at the poles. These models can be interpreted as Coulomb gases on the complex plane with weakly confining potentials, where the associated droplet is the entire complex plane. For these models, we derive precise asymptotic expansions of the free energies, including the constant terms.
Paper Structure (13 sections, 19 theorems, 164 equations, 1 figure)

This paper contains 13 sections, 19 theorems, 164 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Spherical Coulomb gases and weakly confining potential
  3. Main results
  4. Preliminaries and outline of the proof
  5. Stereographic projection
  6. Partition functions and planar (skew)-orthogonal polynomials
  7. Logarithmic potential and critical points
  8. Asymptotic analysis of (skew)-orthogonal norms
  9. Asymptotics of orthogonal norms: Case 1
  10. Asymptotics of orthogonal norms: Case 2 and Case 3
  11. Proof of main results
  12. Proof of Theorem \ref{['Thm_free energy det']}
  13. Proof of Theorem \ref{['Thm_free energy Pfaff']}

Key Result

Lemma 1.1

We have

Figures (1)

  • Figure 1: Illustration of the different asymptotic regimes of orthogonal norms and their associated lemmas, along with a geometric interpretation of these regimes on spherical geometry.

Theorems & Definitions (46)

  • Lemma 1.1
  • Definition 1.2: Background measure
  • Example 1
  • Theorem 1.3: Free energy expansion of the determinantal Coulomb gas on the sphere
  • Remark 1
  • Theorem 1.4: Free energy expansion of the Pfaffian Coulomb gas on the sphere
  • Remark 2
  • Corollary 1.5
  • Remark 3
  • Remark 4
  • ...and 36 more