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Properties of the one-component Coulomb gas on a sphere with two macroscopic external charges

Sung-Soo Byun, Peter J. Forrester, Sampad Lahiry

Abstract

The one-component Coulomb gas on the sphere, consisting on $N$ unit charges interacting via a logarithmic potential, and in the presence of two external charges each of strength proportional to $N$, is considered. There are two spherical caps naturally associated with the external charges, giving rise to two distinct phases depending on them not overlapping (post-critical) or overlapping (pre-critical). The equilibrium measure in the post-critical phase is known from earlier work. We determine the equilibrium measure in the pre-critical phase using a particular conformal map, with the parameters therein specified in terms of a root of a certain fourth order polynomial. This is used to determine the exact form of the electrostatic energy for the pre-critical phase. Using a duality relation from random matrix theory, the partition function for the Coulomb gas at the inverse temperature $β= 2$ can be expanded for large $N$ in the post-critical phase, and in a scaling region of the post and pre-critical boundary. For the pre-critical phase, the duality identity implies a relation between two electrostatic energies, one for the present sphere system, and the other for a certain constrained log-gas relating to the Jacobi unitary ensemble.

Properties of the one-component Coulomb gas on a sphere with two macroscopic external charges

Abstract

The one-component Coulomb gas on the sphere, consisting on unit charges interacting via a logarithmic potential, and in the presence of two external charges each of strength proportional to , is considered. There are two spherical caps naturally associated with the external charges, giving rise to two distinct phases depending on them not overlapping (post-critical) or overlapping (pre-critical). The equilibrium measure in the post-critical phase is known from earlier work. We determine the equilibrium measure in the pre-critical phase using a particular conformal map, with the parameters therein specified in terms of a root of a certain fourth order polynomial. This is used to determine the exact form of the electrostatic energy for the pre-critical phase. Using a duality relation from random matrix theory, the partition function for the Coulomb gas at the inverse temperature can be expanded for large in the post-critical phase, and in a scaling region of the post and pre-critical boundary. For the pre-critical phase, the duality identity implies a relation between two electrostatic energies, one for the present sphere system, and the other for a certain constrained log-gas relating to the Jacobi unitary ensemble.
Paper Structure (17 sections, 12 theorems, 189 equations, 16 figures)

This paper contains 17 sections, 12 theorems, 189 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Outline of the model system and phase transition effect
  3. Relationship to earlier work
  4. Summary of results and outline
  5. Equivalent planar Coulomb gas and the equilibrium measure
  6. Stereographic projection
  7. Equilibrium measure for symmetrically placed charges
  8. Equilibrium measure with the charge $Q_0 N$ at the south pole
  9. Electrostatic energies
  10. Duality identity from random matrix theory and first consequences
  11. The complex (generalised) spherical ensemble
  12. Duality identity
  13. Proof of Proposition \ref{['P1.1']}
  14. Critical regime
  15. Electrostatic consequences of the duality in the pre-critical regime
  16. ...and 2 more sections

Key Result

Proposition 1.1

Consider the partition function associated with (1D) restricted to $\beta = 2$, where $\Omega$ denotes the surface of the sphere and $d \Omega_l = R_{\rm s} \sin \theta_l d \theta_l d \phi$. Require that and fix $\rho_{\rm b} = N(1+Q_0+Q_1)/(4 \pi R_{\rm s}^2)$. We have the large $N$ expansion, independent of $(u_w,v_w)$ to all inverse powers of $N$, for coefficients $\{a_k\}$ expressible in te

Figures (16)

  • Figure 1: $w=w_{ \rm cri } \approx 0.11$
  • Figure 2: $w=0.3$
  • Figure 3: $w=1$
  • Figure 4: $w=3$
  • Figure 5: $w=80$
  • ...and 11 more figures

Theorems & Definitions (24)

  • Proposition 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Remark 2.1
  • Proposition 2.4
  • proof
  • Remark 2.2
  • ...and 14 more