Local Zeta Functions, Complete Graphs and Coulomb Gases

Ponente(s): Victor Manuel Burgos Guerrero

Local zeta functions on graphs are a particular case of the multivariate local zeta functions. Multivariate Local zeta functions are a generalization of the local zeta functions widely studied by Weil, Igusa, among others. We focus on the study of the local zeta function for the complete graph. Using the structure of complete subgraphs and a recursive resolution process of n-2 steps, this local zeta function admits a meromorphic continuation to the [n(n-1)/2]-dimensional complex space, and we obtain explicit expressions for the possible poles. The local zeta function on the complete graph is also known as the Generalized Mehta integral. Mehta integrals have their origin in the study of Gaussian ensembles in random matrix theory and are partition functions of certain gases. In these gases, a logarithmic Coulomb interaction between two charged particles occurs only when an edge connects the sites of these particles. This theory enables us to establish that the partition functions admit a meromorphic continuation in a parameter b (the inverse of the absolute temperature).