  We study a log-gas on a network (a finite, simple graph) confined in a
bounded 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 occurs
only 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 of
multivariate local zeta functions attached to the network and a positive test
function which is determined by the confining potential. The methods and
results of the theory of local zeta functions allow us to establish that the
partition functions admit meromorphic continuations in the parameter \b{eta}
(the inverse of the absolute temperature). We give conditions on the charge
distributions and the confining potential such that the meromorphic
continuations of the partition functions have a pole at a positive value
\b{eta}_{UV}, which implies the existence of phase transitions at finite
temperature. In the case of p-adic fields the meromorphic continuations of the
partition functions are rational functions in the variable p^{-\b{eta}}. We
give an algorithm for computing such rational functions. For this reason, we
can consider the p-adic log-Coulomb gases as exact solvable models. We expect
that all these models for different local fields share common properties, and
that they can be described by a uniform theory.
