Joakim Cronvall: Spectral gaps in the random normal matrix model

Abstract:

We study the two-dimensional Coulomb gas at inverse temperature \(\beta = 2\), what is also known as the random normal matrix model. We are interested in the case when the droplet (the support of the equilibrium measure) consists of several connected components. In particular we study the large \(n\) asymptotics of the partition function \(Z_n \) when the external potential is radial and the droplet consists of several annuli. In this setting we obtain an expansion of \(\log Z_n\) down to constant order. We also find that the asymptotic distribution of smooth linear statistics not only consists of a Gaussian part, but also a discrete distribution describing jumps between components. Based on joint work with Yacin Ameur and Christophe Charlier.