Workshop: The log-gas on a Jordan arc

Speaker
Courteaut, New York University

Abstract
We study a log-gas on an arbitrary Jordan arc in the complex plane, at any positive temperature. We show that as the number of particles tends to infinity, the partition function converges to an expression involving the partition function of the gas on [−1,1], the capacity of the curve to a power depending on the temperature, and the Fredholm determinant of the arc-Grunsky operator. We also obtain an asymptotic formula for the Laplace transform of linear statistics for sufficiently regular test functions, which shows that the centered empirical measure converges to a Gaussian field with explicit asymptotic mean and variance given by the Dirichlet energy of the test function. Based on joint work with Kurt Johansson and Fredrik Viklund.