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arXiv · 2407.06433

Electrostatics on Branching Processes

Christopher D. Sinclair
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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.

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Christopher D. Sinclair. 2024-07-08. Electrostatics on Branching Processes. https://arxiv.org/abs/2407.06433

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