Free energy of the Coulomb gas in the determinantal case on Riemann surfaces
Lucas Bourgoin
TL;DR
This work derives the precise large-$N$ expansion for the log-partition function of a determinantal Coulomb gas on a compact Riemann surface, revealing that the constant term encodes the determinant of the scalar Laplacian and thus connects to Gaussian free-field-type structures. The authors deploy the bosonization formula to relate the magnetic Laplacian determinant, Green functions, and theta data to Arakelov-geometric quantities, enabling an explicit expansion in terms of genus-dependent functionals (Liouville, Mabuchi, Aubin-Yau) and spectral invariants. They provide rigorous asymptotics for the modified partition function and, after integrating out theta, the full partition function, and they establish Gaussian fluctuations for linear statistics with explicit mean and variance formulas. The results confirm the geometric Zabrodin-Wiegmann conjecture in the determinantal case and furnish exact checks for genus zero and one, with comprehensive structure described for higher genus via Arakelov and hyperbolic-geometric data. This bridges random matrix/Coulomb gas theory with Arakelov geometry and spectral theory of Laplacians, offering a detailed toolkit for analyzing two-dimensional Coulomb systems on curved backgrounds.
Abstract
We derive the asymptotic expansion of the partition function of a Coulomb gas system in the determinantal case on compact Riemann surfaces of any genus g. Our main tool is the bosonization formula relating the analytic torsion and geometric quantities including the Green functions appearing in the definition of this partition function. As a result, we prove the geometric version of the Zabrodin-Wiegmann conjecture in the determinantal case.