2023/09/01 by Johansson, Kurt, Viklund, Fredrik · 5 citations
#Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2309.00308
Let D be a Jordan domain of unit capacity. We study the partition function of a planar Coulomb gas in D with a hard wall along η= ∂ D, Zn(D) =\frac 1n!∫Dn∏1≤ k lt; ℓ ≤ n|zk-z_ℓ|2 ∏k=1n d2zk. We are interested in how the geometry of η is reflected in the large n behavior of Zn(D). We prove that η is a Weil-Petersson quasicircle if and only if log Zn(D)= log Zn(\mathbbD) -IL(η)/12 + o(1), n→ ∞, where IL(η) is the Loewner energy of η, \mathbbD is the unit disc, and log Zn(\mathbbD) = log n!/πn. We next consider piecewise analytic η with m corners of interior opening angles παp, p=1,…, m. Our main result is the asymptotic formula log Zn(D)= log Zn(\mathbbD) - \frac 16∑p=1m (αp+\frac 1αp-2 ) log n + o(log n), n→ ∞, which is consistent with physics predictions. The starting point of our analysis is an exact expression for log Zn(D) in terms of a Fredholm determinant involving the truncated Grunsky operator for D. The proof of the main result is based on careful asymptotic analysis of the Grunsky coefficients. As further applications of our method we also study the Loewner energy and the related Fekete-Pommerenke energy, a quantity appearing in the analysis of Fekete points, for equipotentials approximating the boundary of a domain with corners. We formulate several conjectures and open problems.