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).

中文翻译：

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logp-Coulomb Gas具有范数密度的$$ p $$-字段

摘要

本文的主要结果是积分公式

$$ \ int_ {K ^ N} \ rho（x）\ big（\ max_ {i <j} | x_i-x_j | \ big）^ a \ big（\ min_ {i <j} | x_i-x_j | \大）^ b \ prod_ {i <j} | x_i-x_j | ^ {s_ {ij}} \\ || dx | ，$$

其中\（K \）是一个\（P \） -field（即，本地nonarchimedean字段）规范绝对值\（| \ CDOT | \） ，\（N \ GEQ 2 \） ，\（A，B \在\ mathbb {C} \）中，函数\（\ rho：K ^ N \ to \ mathbb {C} \）通过范数\（\ | x \ | = \ max_i具有温和的增长和衰减条件和因子| x_i | \）和\（| dx | \）是\（K ^ N \）的常用Haar度量。该公式是组合数据明确描述的函数的有限和，以及值\（（s_ {ij}）_ {i <j} \ in \ mathbb {C} ^ {\ binom {N} { 2}} \）根据这些数据和参数\（a \），\（b \），\（N \）和\（K \）明确给出积分绝对收敛的值。然后，我们将公式专用于\（s_ {ij} = \ mathfrak {q} _i \ mathfrak {q} _j \ beta \），其中\（\ mathfrak {q} _1，\ mathfrak {q} _2，\ dots， \ mathfrak {q} _N> 0 \）表示\（N \）-对数库仑气体在\（K \）中的电荷，其背景密度为\（\ rho \）和逆温度为\（\ beta \）。从这种专业化中，我们得到混合电荷\（p \）赫塔积分公式的场模拟，以及\（\ max_ {i <j} | x_i-x_j | \）（气体直径）和\（\ min_ {i <j} | x_i-x_j | \）（其粒子之间的最小距离）。