log-Coulomb gas with norm-density in $p$-fields

Abstract

The main result of this paper is a formula for the integral $$\int_{K^N}\rho(x)\big(\max_{i<j}|x_i-x_j|\big)^a\big(\min_{i<j}|x_i-x_j|\big)^b\prod_{i<j}|x_i-x_j|^{s_{ij}}|dx|,$$ where $K$ is a $p$-field (i.e., a nonarchimedean local field) with canonical absolute value $|\cdot|$, $N\geq 2$, $a,b\in\mathbb{C}$, the function $\rho:K^N\to\mathbb{C}$ has mild growth and decay conditions and factors through the norm $\|x\|=\max_i|x_i|$, and $|dx|$ is the usual Haar measure on $K^N$. The formula is a finite sum of functions described explicitly by combinatorial data, and the largest open domain of complex tuples $(s_{ij})_{i<j}$ on which the integral converges absolutely is given explicitly in terms of these data and the parameters $a$, $b$, $N$, and $K$. We then specialize the formula to $s_{ij}=\mathfrak{q}_i\mathfrak{q}_j\beta$, where $\mathfrak{q}_1,\mathfrak{q}_2,\dots,\mathfrak{q}_N>0$ represent the charges of an $N$-particle log-Coulomb gas in $K$ with background density $\rho$ and inverse temperature $\beta$. From this specialization we obtain a mixed-charge $p$-field analogue of Mehta's integral formula, as well as formulas and low-temperature limits for the joint moments of $\max_{i<j}|x_i-x_j|$ (the diameter of the gas) and $\min_{i<j}|x_i-x_j|$ (the minimum distance between its particles).