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 values \((s_{ij})_{i<j}\in\mathbb{C}^{\binom{N}{2}}\) 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).

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Acknowledgments

I would like to sincerely thank my advisor Chris Sinclair for all of the support, advice, and stimulating conversations that first inspired this work and led to many improvements.

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Correspondence to Webster Joe.

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Joe, W. log-Coulomb Gas with Norm-Density in \(p\)-Fields. P-Adic Num Ultrametr Anal Appl 13, 1–43 (2021). https://doi.org/10.1134/S2070046621010015

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Keywords

  • nonarchimedean local field
  • log-Coulomb gas
  • local zeta function
  • set partition
  • tree