log-Coulomb Gases in the Projective Line of a $p$-Field

Abstract

This article extends recent results on log-Coulomb gases in a $p$-field $K$ (i.e., a nonarchimedean local field) to those in its projective line $\mathbb{P}^1(K)$, where the latter is endowed with the $PGL_2$-invariant Borel probability measure and spherical metric. Our first main result is an explicit combinatorial formula for the canonical partition function of log-Coulomb gases in $\mathbb{P}^1(K)$ with arbitrary charge values. Our second main result is called the “$(q+1)$th Power Law”, which relates the grand canonical partition functions for one-component gases in $\mathbb{P}^1(K)$ (where all particles have charge 1) to those in the open and closed unit balls of $K$ in a simple way. The final result is a quadratic recurrence for the canonical partition functions for one-component gases in both unit balls of $K$ and in $\mathbb{P}^1(K)$. In addition to efficient computation of the canonical partition functions, the recurrence provides their “$q\to 1$” limits and “$q\mapsto q^{-1}$” functional equations.