Workshop: Asymptotic expansion of the partition function for β-ensembles with complex potentials

Speaker
Alice Guionnet, ENS Lyon

Abstract
In this work we investigate the asymptotic expansion of integrals of the form \( Z_{V,\Gamma}(N,\beta)= \int_{\Gamma^N} (z_i-z_j)^\beta \exp(-N\sum V(z_i)) dz \) where V is a polynomial with complex coefficients, \(\beta\in 2\mathbb{N}_+\) is an even integer and \(\Gamma\) is an unbounded contour in \(\mathbb{C}\) such that the integral converges. When the equilibrium measure for the associated max-min energy problem is one-cut regular we prove the existence of an asymptotic expansion of \(\log Z_{V,\Gamma}(N,\beta)\) as N goes to infinity. Joint work with Alex Little and Karol Kozlowski.