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

Abstract

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.

arXiv commentFinal Version. Accepted in the Journal of Mathematical Physics

References in corpus (11)

• p-Adic Mathematical Physics
• Motivic Igusa zeta functions
• p-Adic Models of Ultrametric Diffusion Constrained by Hierarchical Energy Landscapes
• p-Adic description of characteristic relaxation in complex systems
• P-adic numbers and replica symmetry breaking
• Replica symmetry breaking related to a general ultrametric space I: replica matrices and functionals
• On $p$-adic Gibbs measures of countable state Potts model on the Cayley tree
• Phase transitions for $P$-adic Potts model on the Cayley tree of order three
• On $P$-adic Ising-Vannimenus model on an arbitrary order Cayley tree
• Continuum limits of sparse coupling patterns
• Non-Archimedean Replicator Dynamics and Eigen's Paradox

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