# Background

$p$-adic Electrostatics

See The $p$-adics and Related Concepts for notation and basics on the $p$-adic numbers, Introduction to Boltzmann Statistics for notation and background on electrostatics, and Basics of Probability for notation and background on probability and $\sigma$-algebras.

Haar measure $\lambda$ on $\mathbb Q_p$, normalized so that $\lambda(\mathbb Z_p) = 1$ is a probability measure on $\mathbb Q_p$. The Borel $\sigma$-algebra $\mathcal B$ on $\mathbb Z_p$ is generated by all balls of radius $\leq 1$ on $\mathbb Z_p$.

In analogy with Two-Dimensional Electrostatics we set an energy functional for $N$ unit charged particles located at $\mathbf x \in \mathbb Z_p^N$ given by $$E_N(\mathbf x) = -\sum_{m<n} \log|x_n – x_m|_p.$$

Questions of interest here are probabilities that prescribed balls each have a prescribed number of particles. See Point Processes for more information. Specifically, for a given Borel set $B \in \mathcal B$ we define the random variable $N_B$ to be the number of particles in $B$. That is, if $X_1, \ldots, X_N$ are $p$-adic valued random variables then $N_B = \# \{ X_1, \ldots, X_N \in B \}.$ Given a collection of balls $\mathbf B = (B_m)$ and vector of non-negative integers $\mathbf n = (n_m)$, then we define the cylinder set $$\{ N_{\mathbf B} = \mathbf n\} = \bigcap_m \{N_{B_m} = n_m\}.$$ If the $B_m$ are pairwise disjoint balls, and there are only finitely many of them, then we call $\{ N_{\mathbf B} = \mathbf n\}$ a basic cylinder set. We define $\mathcal C$ to be the $\sigma$-algebra on $\mathcal Z_p^N$ generated by basic cylinder sets.

Exercise: Show that the set of all basic cyllinder sets forms a $\pi$-system (See Random Variables & Measurable Functions). Show that any cylinder set can be given as a countable union of basic cylinder sets.

Our first goal will be to describe probabilities of basic cylinder sets in the canonical ensemble.