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We study a log-gas on a network (a finite, simple graph) confined in abounded subset of a local field (i.e. R, C, Q_{p} the field of p-adic numbers).In this gas, a log-Coulomb interaction between two charged particles occursonly when the sites of the particles are connected by an edge of the network.The partition functions of such gases turn out to be a particular class ofmultivariate local zeta functions attached to the network and a positive testfunction which is determined by the confining potential. The methods andresults of the theory of local zeta functions allow us to establish that thepartition functions admit meromorphic continuations in the parameter \b{eta}(the inverse of the absolute temperature). We give conditions on the chargedistributions and the confining potential such that the meromorphiccontinuations of the partition functions have a pole at a positive value\b{eta}_{UV}, which implies the existence of phase transitions at finitetemperature. In the case of p-adic fields the meromorphic continuations of thepartition functions are rational functions in the variable p^{-\b{eta}}. Wegive an algorithm for computing such rational functions. For this reason, wecan consider the p-adic log-Coulomb gases as exact solvable models. We expectthat all these models for different local fields share common properties, andthat they can be described by a uniform theory.

Graphs, local zeta functions, Log-Coulomb Gases, and phase transitions at finite temperature