Electrostatics on Branching Processes

TL;DR

This paper studies a probabilistic model of particles on a random tree generated by a branching process, introducing new partition functions that average over the randomness and exhibit algebraic properties.

Contribution

It introduces a novel framework for analyzing electrostatic interactions on random trees using mean partition functions and recursion relations.

Findings

Mean partition functions satisfy recursive relations.

Partition functions exhibit algebraic properties depending on the branching law.

Generalizes known properties from non-random and p-adic cases.

Abstract

We introduce a random probability measure on the profinite completion of the random tree of a branching process and introduce the canonical and grand canonical ensembles of random repelling particles on this random profinite completion at inverse temperature $\beta > 0$. We think of this as a random spatial process of particles in a random tree, and we introduce the notion of the {\em mean} canonical and grand canonical partition functions where in this context `mean' means averaged over the random environment. We give a recursion for these mean partition functions and demonstrate that in certain instances, determined by the law for the branching process, these partition functions as a function of $\beta$ have algebraic properties which generalize those that appear in the non-random and $p$-adic environments.