论文标题
$ p $ fields的木材 - 库仑气体，标准密度
log-Coulomb gas with norm-density in $p$-fields
论文作者
论文摘要
本文的主要结果是积分的公式$ \ int_ {k^n}ρ（x）\ big（\ max_ {i <j} | x_i-x_j | \ big）^a \ big（\ min_ {\ j} | x_i-x_i-x_j | \ big） $ k $是一个$ p $ - 菲尔德（即，非架构的本地田地），具有规范的绝对值$ | \ cdot | $，$ n \ geq 2 $，$ a，$ a，b \ in \ mathbb {c} $，函数$ρ：k^n \ to \ mathbb {c {c} c {c {c} $ \ | x \ | = \ max_i | x_i | $，$ | dx | $是$ k^n $上的通常的HAAR度量。该公式是由组合数据明确描述的有限函数总和，也是复杂元组$（s_ {ij}）的最大开放域_ {i <j} $，绝对在这些数据中明确给出了整体收敛的这些数据，并以这些数据和参数$ a $ a $ a $ a $，$ b $ n $和$ n $和$ k $。然后，我们将公式专门为$ s_ {ij} = \ mathfrak {q} _i \ mathfrak {q}_jβ$，其中$ \ m arterfrak {q} _1，\ mathfrak {q}带有背景密度$ρ$的$ K $和反温度$β$。从这个专业化中，我们获得了Mehta积分公式的混合充电$ p $ field类似物，以及$ \ max_ {i <j} | x_i-x_j | $（天然气的直径）和$ \ \ \ \ \ \ \ \ j {i <j i <j_i-i-x_-x__j | $ x_-x_j | $ x_i-x_j | $ x_-x_j | $ x_i-x_j | $ x_i-x_j | $ x_i-x_j | $ x_i-x_j | $ x_i-x_j | $ x_-x_j | $ x_-x_j | $ x_-x_j | $ x_-x_J |
The main result of this paper is a formula for the integral $$\int_{K^N}ρ(x)\big(\max_{i<j}|x_i-x_j|\big)^a\big(\min_{i<j}|x_i-x_j|\big)^b\prod_{i<j}|x_i-x_j|^{s_{ij}}|dx|,$$ 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 $ρ:K^N\to\mathbb{C}$ 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 $s_{ij}=\mathfrak{q}_i\mathfrak{q}_jβ$, 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 $ρ$ and inverse temperature $β$. 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).
