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

16 Oct 2020 Webster Joe

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 $K$... (read more)

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