Planar Coulomb gas on a Jordan arc at any temperature

Abstract

We study a planar Coulomb gas confined to a sufficiently smooth Jordan arc $\gamma$ in the complex plane, at inverse temperature $\beta > 0$. Let \[\bar{Z}_{n}^\beta(\gamma) = Z_{n}^\beta(\gamma)/\left(2 \textrm{cap}(\gamma)\right)^{\beta n^2/2+(1-\beta/2)n}\] be the normalized partition function. We compute the free energy as the number of particles tends to infinity, including the constant term: \[ \lim_{n\to \infty}\log \frac{\bar{Z}_{n}^\beta(\gamma)}{\bar{Z}_{n}^\beta([-1,1])} = \frac{1}{24}J^A(\gamma) + \frac{1}{8} \left( \sqrt{ \frac{\beta}{2} } - \sqrt{ \frac{2}{\beta}} \right)^2 J^{F}(\gamma). \] Here $\bar{Z}_{n}^\beta([-1,1])$ is an explicit Selberg integral, $J^A(\gamma)$ is half the Loewner energy of a certain Jordan curve associated to $\gamma$ plus a covariance term, and $J^F(\gamma)$ is the Fekete energy, related to the zero temperature limit of the model. We also prove an asymptotic formula for the Laplace transform of linear statistics for sufficiently regular test functions. As a consequence, the centered empirical measure converges to a Gaussian field with explicit asymptotic mean, and asymptotic variance given by the Dirichlet energy of the bounded harmonic extension of the test function outside of the arc. A key tool in our analysis is the arc-Grunsky operator $B$ associated to $\gamma$, reminiscent to but different from the classical Grunsky operator. We derive several basic properties the arc-Grunsky operator, including an estimate analogous to the strengthened Grunsky inequality and the relation to the Dirichlet integral. In our proofs $J^A(\gamma)$ arises from the Fredholm determinant $\det(I+B)$ and a substantial part of the paper is devoted to relating this to the Loewner energy.