Non-Archimedean Coulomb Gases

TL;DR

This paper investigates Coulomb gases in p-adic spaces, establishing equilibrium measures and energy limits, and connecting Coulomb energy with hierarchical spin glass models in a novel non-Archimedean setting.

Contribution

It introduces the study of Coulomb gases over p-adic spaces, proving existence of equilibrium measures and deriving the Gamma-limit of Coulomb energy functional.

Findings

Existence of equilibrium measures in p-adic Coulomb gases.

Explicit computation of equilibrium measure for particles in the unit ball.

Connection between Coulomb energy and hierarchical spin glass Hamiltonian.

Abstract

This article aims to study the Coulomb gas model over the $d$-dimensional $p$-adic space. We establish the existence of equilibria measures and the $\Gamma$-limit for the Coulomb energy functional when the number of configurations tends to infinity. For a cloud of charged particles confined into the unit ball, we compute the equilibrium measure and the minimum of its Coulomb energy functional. In the $p$-adic setting the Coulomb energy is the continuum limit of the minus a hierarchical Hamiltonian attached to a spin glass model with a $p$-adic coupling.

Equations

$$x=x_{-k}{p}^{-k}+x_{-k+1}{p}^{-k+1}+\dots +x_0+x_{1}p+\dots ,\text{ with }{x}_{-k}\ne 0\text{,}$$

$$g_{\alpha }\left(x\right)=\{\begin{array}{ll}||x|{|}_p^{\alpha -d},& \text{ if }\alpha \ne d\\ \ln ||x|{|}_p,& \text{ if }\alpha =d.\end{array}$$

$${\mathbf{D}}^{\alpha }{g}_{\alpha }=-C_{d,\alpha }\delta ,$$

$${\mathcal{E}}_{\alpha }(\mu )={\int }_{{\mathbb{Q}}_p^d}{\int }_{{\mathbb{Q}}_p^d}{g}_{\alpha }\left(x-y\right)d\mu \left(x\right)d\mu \left(y\right)\in \left(-\infty ,+\infty \right].$$

$$I_{\alpha }\left(\mu \right)={\mathcal{E}}_{\alpha }(\mu )+{\int }_{{\mathbb{Q}}_p^d}V\left(x\right)d\mu \left(x\right).$$

$$V(x)=\{\begin{array}{ll}{V}_0,& \text{if }{\Vert x\Vert }_p\le 1\\ +\infty ,& \text{if }{\Vert x\Vert }_p>1,\end{array}$$

$${\int }_{{\mathbb{Q}}_p^d}{g}_{\alpha }\left(x-y\right)d{\mu }_0\left(x\right)+\frac{V}{2}=C,\text{q.e. in the support of }{\mu }_0,$$

$$H_{n,\alpha }(x_1,\dots ,x_n)=\sum _{i\ne j}{g}_{\alpha }\left(x_i-x_j\right)+n\sum _{i}V(i).$$

$$H_{L,l}:=-\sum _{\tilde{x},\tilde{y}\in G_l}{p}^{-2ld}{J}_{\tilde{x}\tilde{y}}\rho \left(\tilde{x}\right)\rho \left(\tilde{y}\right)-\sum _{\tilde{x}\in G_l}{p}^{-ld}\rho \left(\tilde{x}\right)V_0\left(\tilde{x}\right),$$

$$J_{\tilde{x}\tilde{y}}=\{\begin{array}{ll}{\Vert \tilde{x}-\tilde{y}\Vert }_p^{\alpha -d},& \text{if }\tilde{x}\ne \tilde{y},\\ \frac{p^{-l\left(d+\alpha \right)}\left(1-p^{-d}\right)}{1-p^{-\alpha }},& \text{if }\tilde{x}=\tilde{y}.\end{array}$$

$${\left|x\right|}_p=\{\begin{array}{ll}0,& \text{if }x=0\\ p^{-\gamma },& \text{if }x=p^{\gamma }\frac{a}{b},\end{array}$$

$$x=p^{\mathrm{ord}(x)}\sum _{j=0}^{\infty }{x}_j{p}^j,$$

$${\left\{x\right\}}_p=\{\begin{array}{ll}0,& \text{if }x=0\text{ or }\mathrm{ord}(x)\ge 0\\ p^{\mathrm{ord}(x)}{\sum }_{j=0}^{-\mathrm{ord}(x)-1}{x}_j{p}^j,& \text{if }\mathrm{ord}(x)<0.\end{array}$$

$$||x|{|}_p:=\underset{1\le i\le d}{\max }|x_i{|}_p,\text{for }x=(x_1,\dots ,x_d)\in {\mathbb{Q}}_p^d.$$

$$\phi (x+x^{\prime })=\phi (x)\text{for }{x}^{\prime }\in B_{l(x)}^d.$$

$$(\mathcal{F}f)(\xi )={\int }_{{\mathbb{Q}}_p^d}{\chi }_p(\xi \cdot x)f(x)dx,\text{for }\xi \in {\mathbb{Q}}_p^d.$$

$$(\mathcal{F}(\mathcal{F}f))(\xi )=f(-\xi )$$

$$\left(\mathcal{F}\left[W\right],\phi \right)=\left(W,\mathcal{F}\left[\phi \right]\right)\text{for all }\phi \in \mathcal{D}({\mathbb{Q}}_p^N).$$

$$k_{\alpha }(x)=\frac{||x|{|}_p^{\alpha -d}}{{\Gamma }_p^{\left(d\right)}\left(\alpha \right)},\alpha \in \mathbb{R}\setminus \left\{0,d\right\},x\in {\mathbb{Q}}_p^d,$$

$$\begin{array}{rl}\left(k_{\alpha },\phi \right)& =\frac{1-p^{-d}}{1-p^{\alpha -d}}\phi (0)+\frac{1-p^{-\alpha }}{1-p^{\alpha -d}}{\int }_{||x|{|}_p>1}||x|{|}_p^{\alpha -d}\phi (x)dx\\ & +\frac{1-p^{-\alpha }}{1-p^{\alpha -d}}{\int }_{||x|{|}_p\le 1}||x|{|}_p^{\alpha -d}(\phi (x)-\phi (0))dx.\end{array}$$

$$\left(k_0,\phi \right):=\underset{\alpha \to 0}{\lim }\left(k_{\alpha },\phi \right)=\phi (0),$$

$$\left(k_{-\alpha },\phi \right)=\frac{1-p^{\alpha }}{1-p^{-\alpha -d}}{\int }_{{\mathbb{Q}}_p^n}||x|{|}_p^{-\alpha -d}(\phi (x)-\phi (0))dx.$$

$${\mathbf{D}}^{\alpha }\phi (x)={\mathcal{F}}_{\xi \to x}^{-1}\left(||\xi |{|}_p^{\alpha }{\mathcal{F}}_{x\to \xi }\phi \right),\text{for }\phi \in \mathcal{D}.$$

$${\mathbf{D}}^{\alpha }\phi \left(x\right)=\left(k_{-\alpha }\ast \phi \right)\left(x\right)=\frac{1-p^{\alpha }}{1-p^{-\alpha -d}}{\int }_{{\mathbb{Q}}_p^d}||y|{|}_p^{-\alpha -d}(\phi (x-y)-\phi (x))dy.$$

$${\int }_{||x|{|}_p\ge 1}||x|{|}_p^{-\alpha -d}|\phi (x)|dx<\infty .$$

$$\begin{array}[c]{ccc}\operatorname{Dom}(\boldsymbol{D}^{\alpha})&\rightarrow&\mathcal{D}^{\prime}\vskip 3.0pt plus 1.0pt minus 1.0pt\\ T&\rightarrow&\boldsymbol{D}^{\alpha}T,\end{array}$$

$$\frac{\partial u(x,t)}{\partial t}+{\mathbf{D}}^{\alpha }u(x,t)=0,x\in {\mathbb{Q}}_p^d,t>0.$$

$$u(x,t)={\int }_{{\mathbb{Q}}_p^d}Z(x-y,t)\phi (x)dx,$$

$$Z(x,t):={\int }_{{\mathbb{Q}}_p^d}{\chi }_p(-x\cdot \xi )e^{-t||\xi |{|}_p^{\alpha }}d\xi \text{for }t>0,$$

$${\mathbf{D}}^{\alpha }u(x)=\phi \left(x\right),\phi \in \mathcal{D}.$$

Full text

The first named author was partially supported by Conacyt Grant No. 222478. The second named author was partially supported by Conacyt Grant No. 250845.

1. 1 Introduction

2. 2 Basic aspects of the $p$**-**adic analysis

3. 2.1 The field of $p$-adic numbers

4. 2.2 Some function spaces

5. 2.3 Fourier transform

6. 3 The Taibleson operator

7. 3.1 $p$-adic heat equations

8. 3.2 Fundamental solutions

9. 4 Some technical results

10. 5 The $p$-adic Coulomb gas

11. 5.1 $\Gamma$-convergence

12. 5.2 $\Gamma$-convergence of the $p$-adic Coulomb gas Hamiltonian

13. 5.3 Minimizing the mean-field energy via potential theory

14. 6 The Coulomb gas confined into the unit ball

15. 7 Proof of the $\Gamma$-convergence and some consequences

16. 7.1 Proof of Theorem 1

17. 7.2 Some further results

18. 7.3 Continuum limits of hierarchical models

1. Introduction

In this article we initiate the study of Coulomb gases on the $d$-dimensional $p$-adic space $\mathbb{Q}_{p}^{d}$. More precisely, we give $p$-adic counterparts of the existence and characterization of the equilibrium measure, see Theorem 2, the $\Gamma$-convergence of the Coulomb energy functional, see Theorem 1, and the convergence of the minimizers of this functional, see Theorem 3. For the classical counterparts the reader may consult, for instance, Serfaty’s book [33, Proposition 2.8, Theorems 2.1, 2.2].

From a mathematical perspective, the results presented here are framed in the probability and potential theory over ultrametric spaces. Probability over ultrametric spaces has been studied extensively during the last thirty years, see e.g. [6], [7], [10] and the references therein, due, among several things, to the emergence of the use of ultrametric spaces in physical models, see e.g. [20], [30], [38], [40] and the references therein. On the other hand, the study of potential theory over locally compact Abelian groups, see e.g. [8], and over metric spaces, see e.g. [1], [9], is a classical matter.

From a physical perspective, the study of models over ultrametric spaces started in the middle 80s with the works of Frauenfelder, Parisi, Stein, among others, see e.g. [10], [14], [26], [30], see also [3], [4], [19], [20], [40], and the references therein. The key idea is that the space of states of certain physical systems have a natural structure of ultrametric space. An ultrametric space $(M,d)$ is a metric space $M$ with a distance satisfying the strong triangle inequality $d(A,B)\leq\max\left\{d\left(A,C\right),d\left(B,C\right)\right\}$ for any three points $A$, $B$, $C$ in $M$.

The Ising models over ultrametric spaces have been studied intensively, see e.g. [11], [16], [18], [19], [21], [25], [27], [28], [29], [34] and the references therein,  motivated, among several things, by the hierarchical Ising model introduced in [11]. The hierarchical Hamiltonian introduced by Dyson in [11] can be naturally studied in $p$-adic spaces, see e.g. [21], [16]. These Hamiltonians are self-similar with respect to suitable scale groups.

A $p$-adic number is a series of the form

[TABLE]

where $p$ denotes a fixed prime number, and the $x_{j}$s  are $p$-adic digits, i.e. numbers in the set $\left\{0,1,\ldots,p-1\right\}$. There are natural field operations, sum and multiplication, on series of form (1.1). The set of all possible $p$-adic numbers constitutes the field of $p$-adic numbers $\mathbb{Q}_{p}$. There is also a natural norm in $\mathbb{Q}_{p}$ defined as $\left|x\right|_{p}=p^{k}$, for a nonzero $p$-adic number $x$ of the form (1.1). The field of $p$-adic numbers with the distance induced by $\left|\cdot\right|_{p}$ is a complete ultrametric space. The ultrametric property refers to the fact that $\left|x-y\right|_{p}\leq\max\left\{\left|x-z\right|_{p},\left|z-y\right|_{p}\right\}$ for any $x$, $y$, $z$ in $\mathbb{Q}_{p}$. As a topological space, $\left(\mathbb{Q}_{p},\left|\cdot\right|_{p}\right)$ is completely disconnected, i.e. the connected components are points. The field of $p$-adic numbers has a fractal structure, see e.g. [2], [38]. We extend the $p$-adic norm to $\mathbb{Q}_{p}^{d}$, by taking $\left\|\left(x_{1},\ldots,x_{d}\right)\right\|=\max_{i}\left|x_{i}\right|_{p}$.

For $\alpha>0$, the $d$-dimensional $p$-adic Coulomb kernel is defined as

[TABLE]

This kernel is similar to the classic one, however, in the $p$-adic setting, we have a family of kernels parametrized by $\alpha>0$. In this article we consider only the kernels $\frac{1}{||x||_{p}^{d-\alpha}}$, with $d>\alpha$. The function $g_{\alpha}\left(x\right)$ is the fundamental solution of a ‘$p$-adic Poisson’s equation.’ In the $p$-adic framework, there are infinitely many ‘Laplacians’. By a Laplacian we mean an operator $A$ such that the semigroup generated by $-A$ is Markovian. We pick the simplest possible Laplacian in dimension $d$, the Taibleson operator $\boldsymbol{D}^{\alpha}$, $\alpha>0$, which is a pseudodifferential operator defined as $\mathcal{F}\left(\boldsymbol{D}^{\alpha}\varphi\right)=||\xi||_{p}^{\alpha}\mathcal{F}\varphi$, where $\mathcal{F}$ denotes the Fourier transform. Notice that here $\alpha$ is an arbitrary positive number, while in the classical case, similar operators exist only if $\alpha\in\left(0,2\right]$. If we consider $g_{\alpha}\left(x\right)$ as distribution, then

[TABLE]

where $\delta$ denotes the Dirac distribution at the origin and $C_{d,\alpha}$ is a constant, see Section 3.

Let $\mathcal{P}(\mathbb{Q}_{p}^{d})$ be the space of probability measures on $\mathbb{Q}_{p}^{d}$. The Coulomb energy of the measure $\mu$ is defined as

[TABLE]

Now we introduce and admissible potential $V:\mathbb{Q}_{p}^{d}\rightarrow\left(-\infty,+\infty\right]$ satisfying the standard conditions. For this potential we consider the Coulomb energy functional

[TABLE]

We show the existence of a unique minimizer $\mu_{0}$ ($\min_{\mu\in\mathcal{P}(\mathbb{Q}_{p}^{d})}\left\{I_{\alpha}\left(\mu\right)\right\}=I_{\alpha}\left(\mu_{0}\right)$) called the equilibrium measure, see Theorem 2.

Since $\left(\mathbb{Q}_{p}^{d},\left\|\cdot\right\|_{p}\right)$ is a Polish space, we can use classical probability techniques to establish Theorem 2. This result is a $p$-adic version of the Frostman theorem, see e.g. [33, Theorem 2.1]. In the case $V\equiv 0$, this result is well-known in the context of locally compact Abelian groups, see e.g. [8, Theorem 16.22].

At first sight, Theorem 2 is not very different of the classical one. However, there are several important differences, among them, suitable locally constant functions are admissible potentials; second, the ultrametric topology of $\mathbb{Q}_{p}^{d},$ imposes new restrictions on the equilibria measures; and third, operator $\boldsymbol{D}^{\alpha}$ is non local. This last fact makes the computation of the equilibrium measures very difficult.

Consider the potential

[TABLE]

where $V_{0}>0$. The energy functional $I(\mu)$ corresponds to a cloud of charged particles confined into the unit ball. In Proposition 3, we compute the equilibrium measure $\mu_{0}$ for $I(\mu)$.  In the classical approach, one applies the Laplacian to an equality of the form

[TABLE]

see Theorem 2, to obtain a formula for $\mu_{0}\left(x\right)$ in an open set contained in the support of $\mu_{0}$. In the $p$-adic case, this approach is not possible due to the fact that the operator $\boldsymbol{D}^{\alpha}$ is non local, see Section 6 and Proposition 3.

The Hamiltonian $H_{n,\alpha}(x_{1},\ldots,x_{n})$ of the Coulomb gas corresponding to the configuration $x_{1},\ldots,\allowbreak x_{n}\in\mathbb{Q}_{p}^{d}$ is defined as

[TABLE]

Under the assumptions that $V$ is continuous and bounded from below, and that $g_{\alpha}(x)=\frac{1}{\left\|x\right\|_{p}^{d-\alpha}}$, with $d>\alpha>0$, we show that $\frac{1}{n^{2}}H_{n,\alpha}$  $\Gamma$-converges to $I_{\alpha}\left(\mu\right)$, see Theorem 1, i.e. $I_{\alpha}\left(\mu\right)$ is the mean-field energy of $\frac{1}{n^{2}}H_{n,\alpha}$.

We also consider the configurations $x_{1},\ldots,x_{n}$ minimizing the corresponding Hamiltonians $H_{n,\alpha}$, $n\in\mathbb{N}$ and show that $\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}\rightarrow\mu_{0}$ in the weak sense of probability measures, and that $\lim_{n\rightarrow+\infty}\frac{1}{n^{2}}H_{n,\alpha}\left(x_{1},\ldots,x_{n}\right)=I(\mu_{0})$, see Theorem 3.

For $L\in Z$ fixed and $l\geq-L$, set $G_{l}=p^{-L}\mathbb{Z}_{p}^{d}/p^{l}\mathbb{Z}_{p}^{d}$, where $\mathbb{Z}_{p}^{d}$ is the $d$-dimensional unit ball. Then $G_{l}$ is naturally a finite ultrametric space. Consider the Hamiltonian

[TABLE]

where $\rho$ and $V_{0}$ are real-valued functions and the coupling $J_{\widetilde{x}\,\widetilde{y}}$ is given by

[TABLE]

Then $H_{L,l}$ is the Hamiltonian of a spin glass model with $p$-adic coupling, see [16, Section C] . Under general conditions about functions $\rho$ and $V_{0}$,   we obtain that $-\lim_{l\rightarrow\infty}H_{L},_{l}$ agrees with the Coulomb energy attached to the measure $\rho\,dx$ and a potential which is infinite outside the ball $p^{-L}\mathbb{Z}_{p}^{d}$, and that agrees with the function $V_{0}$ inside the ball $p^{-L}\mathbb{Z}_{p}^{d}$, see Section 7.3.

The Coulomb gas model is related with several relevant matters, among them, random matrices and the obstacle problem, see e.g. [33, Chapter 2]. The theory of $p$-adic random matrices is not fully developed, but it is connected with relevant number-theoretic matters, see e.g. [12], see also [13]. We expect that the $p$-adic Coulomb gas model will be useful in the study of $p$-adic random matrices. On the other hand, discrete versions of the obstacle problem play a central role in the study of sandpile models, see e.g. [22]. Sandpile models have been studied on infinite trees, see e.g. [24], which are ultrametric spaces. We expect that $p$-adic versions of the obstacle problem will play a central role in the construction of $p$-adic counterparts of sandpile models. Finally, all the results presented in this work are still valid if we replace $\mathbb{Q}_{p}$ by $\mathbb{F}_{p}((t))$, the field of formal Laurent power series with coefficients in the finite field $\mathbb{F}_{p}$ with $p$ elements. In the recent preprint [35], Sinclair and Vaaler study the partition function for a $p$-adic Coulomb gas confined into the unit ball in the case $g_{d}\left(x\right)=\ln\left\|x\right\|_{p}$. This partition function is a local zeta function attached to the Vandermonde determinant.

2. Basic aspects of the $p$**-**adic analysis

In this section we collect some basic results about $p$-adic analysis that will be used in the article. For an in-depth review of the $p$-adic analysis the reader may consult [2], [36], [38].

2.1. The field of $p$-adic numbers

Along this article $p$ will denote a prime number. The field of $p-$adic numbers $\mathbb{Q}_{p}$ is defined as the completion of the field of rational numbers $\mathbb{Q}$ with respect to the $p-$adic norm $|\cdot|_{p}$, which is defined as

[TABLE]

where $a$ and $b$ are integers coprime with $p$. The integer $\gamma=:\operatorname{ord}(x)$, with $\operatorname{ord}(0):=+\infty$, is called the* $p-$adic order of* $x$.

Any $p-$adic number $x\neq 0$ has a unique expansion of the form

[TABLE]

where $x_{j}\in\{0,\dots,p-1\}$ and $x_{0}\neq 0$. By using this expansion, we define the fractional part of $x\in\mathbb{Q}_{p}$, denoted $\{x\}_{p}$, as the rational number

[TABLE]

In addition, any non-zero $p-$adic number can be represented uniquely as $x=p^{\operatorname{ord}(x)}\operatorname{ac}\left(x\right)$ where $\operatorname{ac}\left(x\right)=\sum_{j=0}^{\infty}x_{j}p^{j}$, $x_{0}\neq 0$, is called the angular component of $x$. Notice that $\left|\operatorname{ac}\left(x\right)\right|_{p}=1$.

We extend the $p-$adic norm to $\mathbb{Q}_{p}^{d}$ by taking

[TABLE]

We define $\operatorname{ord}(x)=\min_{1\leq i\leq d}\{\operatorname{ord}(x_{i})\}$, then $||x||_{p}=p^{-\operatorname{ord}(x)}$. The metric space $\left(\mathbb{Q}_{p}^{d},||\cdot||_{p}\right)$ is a separable complete ultrametric space. For $r\in\mathbb{Z}$, denote by $B_{r}^{d}(a)=\{x\in\mathbb{Q}_{p}^{d};||x-a||_{p}\leq p^{r}\}$ the ball of radius $p^{r}$ with center at $a=(a_{1},\dots,a_{d})\in\mathbb{Q}_{p}^{d}$, and take $B_{r}^{d}:=B_{r}^{d}(0)$. Note that $B_{r}^{d}(a)=B_{r}(a_{1})\times\cdots\times B_{r}(a_{d})$, where $B_{r}(a_{i}):=\{x\in\mathbb{Q}_{p};|x_{i}-a_{i}|_{p}\leq p^{r}\}$ is the one-dimensional ball of radius $p^{r}$ with center at $a_{i}\in\mathbb{Q}_{p}$. The ball $B_{0}^{d}$ equals to the product of $d$ copies of $B_{0}=\mathbb{Z}_{p}$, the ring of $p-$adic integers of $\mathbb{Q}_{p}$. We also denote by $S_{r}^{d}(a)=\{x\in\mathbb{Q}_{p}^{d};||x-a||_{p}=p^{r}\}$ the sphere of radius $p^{r}$ with center at $a=(a_{1},\dots,a_{d})\in\mathbb{Q}_{p}^{d}$, and take $S_{r}^{d}:=S_{r}^{d}(0)$. We notice that $S_{0}^{1}=\mathbb{Z}_{p}^{\times}$ (the group of units of $\mathbb{Z}_{p}$), but $\left(\mathbb{Z}_{p}^{\times}\right)^{d}\subsetneq S_{0}^{d}$. The balls and spheres are both open and closed subsets in $\mathbb{Q}_{p}^{d}$. In addition, two balls in $\mathbb{Q}_{p}^{d}$ are either disjoint or one is contained in the other.

As a topological space $\left(\mathbb{Q}_{p}^{d},||\cdot||_{p}\right)$ is totally disconnected, i.e. the only connected  subsets of $\mathbb{Q}_{p}^{d}$ are the empty set and the points. A subset of $\mathbb{Q}_{p}^{d}$ is compact if and only if it is closed and bounded in $\mathbb{Q}_{p}^{d}$, see e.g. [38, Section 1.3], or [2, Section 1.8]. The balls and spheres are compact subsets. Thus $\left(\mathbb{Q}_{p}^{d},||\cdot||_{p}\right)$ is a locally compact topological space.

We will use $\Omega\left(p^{-r}||x-a||_{p}\right)$ to denote the characteristic function of the ball $B_{r}^{d}(a)$. We will use the notation $1_{A}$ for the characteristic function of a set $A$. Along the article $dx$ will denote a Haar measure on $\left(\mathbb{Q}_{p}^{d},+\right)$ normalized so that $\int_{\mathbb{Z}_{p}^{d}}dx=1.$

2.2. Some function spaces

A complex-valued function $\varphi$ defined on $\mathbb{Q}_{p}^{d}$ is called locally constant if for any $x\in\mathbb{Q}_{p}^{d}$ there exist an integer $l(x)\in\mathbb{Z}$ such that

[TABLE]

A function $\varphi:\mathbb{Q}_{p}^{d}\rightarrow\mathbb{C}$ is called a Bruhat-Schwartz function, or a test function, if it is locally constant with compact support. In this case, there exists $l\in\mathbb{Z}$, independent of $x$, such that (2.1) holds. The largest of such numbers $l=l\left(\varphi\right)$ is called the index of local constancy of $\varphi$. The $\mathbb{C}$-vector space of Bruhat-Schwartz functions is denoted by $\mathcal{D}:=\mathcal{D}(\mathbb{Q}_{p}^{d})$. We will denote by $\mathcal{D}_{\mathbb{R}}:=\mathcal{D}_{\mathbb{R}}(\mathbb{Q}_{p}^{d})$, the $\mathbb{R}$-vector space of test functions. The convergence in $\mathcal{D}$ is defined in the following way: $\varphi_{k}\rightarrow 0$, $k\rightarrow\infty$, in $\mathcal{D}$ if and only if

• (i)

all the $\varphi_{k}$s are supported in a ball $B_{N}^{d}$ and have indices of local constancy $l(\varphi_{k})\geq l$, with $N$ and $l$ independent of $k$;

• (ii)

$\varphi_{k}\rightarrow 0$ uniformly in $\mathbb{Q}_{p}^{d}$.

Let $\mathcal{D}^{\prime}:=\mathcal{D}^{\prime}(\mathbb{Q}_{p}^{d})$ denote the set of all continuous functionals (distributions) on $\mathcal{D}$. We will denote by $\mathcal{D}_{\mathbb{R}}^{\prime}:=\mathcal{D}_{\mathbb{R}}^{\prime}(\mathbb{Q}_{p}^{d})$ the $\mathbb{R}$-vector space of distributions. The convergence in $\mathcal{D}^{\prime}$ is the weak convergence: $T_{k}\rightarrow 0$, $k\rightarrow\infty$, in $\mathcal{D}^{\prime}$ if $\left(T_{k},\varphi\right)\rightarrow 0$, $k\rightarrow\infty$, for any $\varphi\in\mathcal{D}$.

Given $\rho\in[1,\infty)$ and an open subset $U\subset\mathbb{Q}_{p}^{d}$, we denote by $L^{\rho}:=L^{\rho}\left(U\right)$ the $\mathbb{C}$-vector space of all the complex valued functions $g$ defined on $U$ satisfying $\left\|g\right\|_{\rho}=\left\{\int_{U}\left|g\left(x\right)\right|^{\rho}dx\right\}^{\frac{1}{\rho}}<\infty$, and $L^{\infty}\allowbreak:=L^{\infty}\left(U\right)$ denotes the $\mathbb{C}-$vector space of all the complex valued functions $g$ defined in $U$ such that the essential supremum of $|g|$ is bounded. The corresponding $\mathbb{R}$-vector spaces are denoted as $L_{\mathbb{R}}^{\rho}\allowbreak:=L_{\mathbb{R}}^{\rho}\left(U\right)$, $1\leq\rho\leq\infty$.

Let $U$ be an open subset of $\mathbb{Q}_{p}^{d}$, we denote by $\mathcal{D}(U)$ the $\mathbb{C}$-vector space of all test functions from $\mathcal{D}(\mathbb{Q}_{p}^{d})$ with supports in $U$. For each $\rho\in[1,\infty)$, $\mathcal{D}(U)$ is dense in $L^{\rho}\left(U\right)$, see e.g. [2, Proposition 4.3.3].

2.3. Fourier transform

Set $\chi_{p}(y):=\exp(2\pi i\{y\}_{p})$ for $y\in\mathbb{Q}_{p}$. The map $\chi_{p}(\cdot)$ is an additive character on $\mathbb{Q}_{p}$, i.e. a continuous map from $\left(\mathbb{Q}_{p},+\right)$ into $S$ (the unit circle considered as multiplicative group) satisfying $\chi_{p}(x_{0}+x_{1})=\chi_{p}(x_{0})\chi_{p}(x_{1})$, $x_{0},x_{1}\in\mathbb{Q}_{p}$. The additive characters of $\mathbb{Q}_{p}$ form an Abelian group which is isomorphic to $\left(\mathbb{Q}_{p},+\right)$, the isomorphism is given by $\xi\rightarrow\chi_{p}(\xi x)$, see e.g. [2, Section 2.3].

Given $x=(x_{1},\dots,x_{d}),$ $\xi=(\xi_{1},\dots,\xi_{d})\in\mathbb{Q}_{p}^{d}$, we set $x\cdot\xi:=\sum_{j=1}^{d}x_{j}\xi_{j}$. If $f\in L^{1}$, its Fourier transform is defined by

[TABLE]

We will also use the notation $\mathcal{F}_{x\rightarrow\xi}f$ and $\widehat{f}$ for the Fourier transform of $f$. The Fourier transform can be extended as a unitary operator onto $L^{2}$, satisfying

[TABLE]

for every $f\in L^{2}$, see e.g. [2, Sections 2.3 and 4.8] and [36, Chapter III, Section 2].

The Fourier transform $\mathcal{F}\left[W\right]$ of a distribution $W\in\mathcal{D}^{\prime}\left(\mathbb{Q}_{p}^{d}\right)$ is defined by

[TABLE]

The Fourier transform $W\rightarrow\mathcal{F}\left[W\right]$ is a linear isomorphism from $\mathcal{D}^{\prime}\left(\mathbb{Q}_{p}^{N}\right)$ onto itself. Furthermore, $W\left(\xi\right)=\mathcal{F}\left[\mathcal{F}\left[W\right]\left(-\xi\right)\right]$. We also use the notation $\mathcal{F}_{x\rightarrow\xi}W$ and $\widehat{W}$ for the Fourier transform of $W.$

3. The Taibleson operator

We set $\Gamma_{p}^{\left(d\right)}(\alpha):=\frac{1-p^{\alpha-d}}{1-p^{-\alpha}}$, $\alpha\neq 0$. The function

[TABLE]

is called the multi-dimensional Riesz kernel; it determines a distribution on $\mathcal{D}(\mathbb{Q}_{p}^{d})$ as follows. If $\alpha\not\in\{0,d\}$, and $\varphi\in\mathcal{D}(\mathbb{Q}_{p}^{d})$, then

[TABLE]

Then $k_{\alpha}\in\mathcal{D}^{\prime}(\mathbb{Q}_{p}^{d})$, for $\mathbb{R\setminus}\left\{0,d\right\}$. In the case $\alpha=0$, by passing to the limit in (3.1), we obtain

[TABLE]

i.e., $k_{0}(x)=\delta\left(x\right)$, the Dirac delta distribution, and therefore $k_{\alpha}\in\mathcal{D}^{\prime}(\mathbb{Q}_{p}^{d})$, for $\mathbb{R\setminus}\left\{d\right\}$.

It follows from (3.1) that for $\alpha>0$,

[TABLE]

Definition 1**.**

The Taibleson pseudodifferential operator $\boldsymbol{D}^{\alpha}$, $\alpha>0$, is defined as

[TABLE]

This operator was introduced in [36], see also [31] and [2, Chapter 9]. The Taibleson operator coincides with the Vladimirov operator in dimension one.

From the fact that $\left(\mathcal{F}k_{\alpha}\right)\left(x\right)$, with $\alpha\neq d$, equals to $||x||_{p}^{-\alpha}$ in $\mathcal{D}^{\prime}$, see e.g. [36, Chap. III, Theorem 4.5], and (3.2), we have

[TABLE]

The right-hand side of (3.3) makes sense for a wider class of functions, for example, for locally constant functions $\varphi$ satisfying

[TABLE]

Consequently, we may assume that the constant functions are contained in the domain of $\boldsymbol{D}^{\alpha}$, and that $\boldsymbol{D}^{\alpha}\varphi=0$, for any constant function $\varphi$. Later on, we will work with the following extension of $\boldsymbol{D}^{\alpha}$:

[TABLE]

where $\operatorname{Dom}(\boldsymbol{D}^{\alpha}):=\left\{T\in\mathcal{D}^{\prime};||x||_{p}^{\alpha}\mathcal{F}\left(T\right)\in\mathcal{D}^{\prime}\right\}$, and $\boldsymbol{D}^{\alpha}T=\mathcal{F}^{-1}(||x||_{p}^{\alpha}\mathcal{F}\left(T\right))$. Notice that for this operator, the formula $\boldsymbol{D}^{\alpha}T=k_{-\alpha}\ast T$ holds.

3.1. $p$-adic heat equations

In this article the Taibleson operator $\boldsymbol{D}^{\alpha}$ will be considered as a $p$-adic analog of the Laplacian $\Delta:=\frac{\partial^{2}}{\partial x_{1}^{2}}+\cdots+\frac{\partial^{2}}{\partial x_{d}^{2}}$ in $\mathbb{R}^{d}$. To explain this analogy, we use the ‘$p$-adic heat equation,’ which is defined as

[TABLE]

The analogy with the classical heat equation comes from the fact that the solution of the initial value problem attached to (3.4) with initial datum $u(x,0)=\varphi(x)\in\mathcal{D}_{\mathbb{R}}$ is given by

[TABLE]

where

[TABLE]

is the $p$-adic heat kernel. $Z\left(x,t\right)$ is a transition density of a time and space homogeneous Markov process which is bounded, right continuous and has no discontinuities other than jumps, cf. [40, Theorem 16].

The family of ‘$p$-adic Laplacians’ is very large, see e.g. [2, Chapter 9], [20, Chapter 12], [17, Chapter 4], [37], [40, Chapter 2] and the references therein. We pick the Taibleson operator due to the fact that the corresponding fundamental solutions are well-known.

3.2. Fundamental solutions

The $p$-adic analog of the electrostatic equation is

[TABLE]

A fundamental solution of (3.5) is a distribution $G_{\alpha}$ such that $u=G_{\alpha}\ast\varphi$ is a solution of (3.5) in $\mathcal{D}^{\prime}$.

Proposition 1** ([32, Theorem 13]).**

A fundamental solution for (3.5) is given by

[TABLE]

By using that $\boldsymbol{D}^{\alpha}{\LARGE\cdot}=k_{-\alpha}\ast{\LARGE\cdot}$, and that $\boldsymbol{D}^{\alpha}\left(G_{\alpha}\ast\varphi\right)=\varphi$ in $\mathcal{D}^{\prime}$, for any $\varphi\in\mathcal{D}$, we have

[TABLE]

for any test function $\varphi$, and consequently $k_{-\alpha}\ast G_{\alpha}=\delta$, i.e.

[TABLE]

To allow an easy comparison with the literature on Coulomb gases, we set:

[TABLE]

then

[TABLE]

Notice that in the Archimedean case $\alpha=2$, while in the non-Archimedean case, we have a family of Green functions depending on the parameter $\alpha$. In addition, in the $p$-adic case, the potentials $||x||_{p}^{\alpha-d}$, $\ln||x||_{p}$ occur in all the dimensions.

From now on, we assume that $g_{\alpha}\left(x\right)=\frac{1}{||x||_{p}^{d-\alpha}}$ with $d>\alpha>0$.

4. Some technical results

Lemma 1**.**

For $x$, $y\in\mathbb{Q}_{p}^{d}$, with $x\neq y$, and $\alpha>0$, with $d>\alpha$, we set

[TABLE]

Then

[TABLE]

Proof.

The announced formula is proved by a sequence of changes of variables. By changing variables as $z\rightarrow t^{-1}w+y$, $t\rightarrow t$ (then $dz\,dt\rightarrow\left|t\right|_{p}^{-d}dw\,dt$) in (4.1), and using that $\Omega\ast\Omega=\Omega$, we get

[TABLE]

Finally, we change the variables as $t\left\|\left(x-y\right)\right\|_{p}^{-1}\rightarrow s$ (then $dt\rightarrow\left\|\left(x-y\right)\right\|_{p}^{-1}ds$) to obtain

[TABLE]

Let $\mu$, $\nu$ be signed Radon measures on $\mathbb{Q}_{p}^{d}$. We set, for $d>\alpha$,

[TABLE]

Proposition 2**.**

If $\mathcal{E}_{\alpha}(\left|\mu\right|,\left|\mu\right|)<+\infty$, then

[TABLE]

The equality in (4.2) holds if and only if $\mu=0$. Moreover, if $\mathcal{E}_{\alpha}(\left|\nu\right|,\left|\nu\right|)<+\infty$, we have the inequality

[TABLE]

with the equality for $\nu\neq 0$ if and only if $\mu=c\nu$ for some constant $c$. The map $\mu\rightarrow\mathcal{E}_{\alpha}(\mu,\mu)$ is strictly convex, i.e. when $\mu\neq\nu$ and $0<\lambda<1$,

[TABLE]

Proof.

By applying Lemma 1 and Fubini’s theorem, here we use the hypothesis $\mathcal{E}_{\alpha}(\left|\mu\right|,\left|\mu\right|)<\infty$, we have

[TABLE]

where

[TABLE]

Now, assume that $\mathcal{E}_{\alpha}(\mu,\mu)=0$. Then, it follows from (4.5) that $\mu\left(z\right)\ast 1_{B_{\operatorname{ord}(t)}^{d}}\left(z\right)\equiv 0$ (which is a locally constant function) for almost all $t\in\mathbb{Q}_{p}$. This last function is locally constant in $z$ (for almost all $t)$, and its Fourier transform $\widehat{\mu\ast 1_{B_{\operatorname{ord}(t)}^{d}}}\left(\xi\right)=p^{d\operatorname{ord}(t)}\widehat{\mu}\left(\xi\right)\cdot\Omega\left(\left\|p^{-\operatorname{ord}(t)}\xi\right\|\right)$, consequently, $\widehat{\mu}\left(\xi\right)=0$ for any $\left\|\xi\right\|_{p}\leq\left|t\right|_{p}$. And since $t\in\mathbb{Q}_{p}\smallsetminus\left\{0\right\}$ is arbitrary, $\widehat{\mu}=0$, and thus $\mu=0$.

Inequality (4.3) is proved by considering $\mathcal{E}_{\alpha}(\mu-\lambda\nu,\mu-\lambda\nu)\geq 0$, with $\lambda=\frac{\mathcal{E}_{\alpha}(\mu,\nu)}{\mathcal{E}_{\alpha}(\nu,\nu)}$.

To prove inequality (4.4), we use that right hand side minus the left hand side is equal to

[TABLE]

This proposition is the $p$-adic counterpart of Theorem 9.8 in [23].

5. The $p$-adic Coulomb gas

The Hamiltonian of the $p$-adic Coulomb gas is defined as

[TABLE]

where $x_{1},\ldots,x_{n}\in\mathbb{Q}_{p}^{d}$ and $V:\mathbb{Q}_{p}^{d}\rightarrow\mathbb{R}$. In this article we only consider the case $g_{\alpha}\left(x\right)=\frac{1}{||x||_{p}^{d-\alpha}}$, with $d>\alpha$.

5.1. $\Gamma$-convergence

Definition 2**.**

We say that a sequence $\left\{F_{n}\right\}_{n\in\mathbb{N}}$ of functions, on a metric space $X$, $\Gamma$-converges to a function $F:X\rightarrow\left(-\infty,+\infty\right]$ if the following two inequalities hold:

($\Gamma-\lim\inf$) if $x_{n}\rightarrow x$ in $X$, then $\liminf_{n\rightarrow+\infty}F_{n}\left(x_{n}\right)\geq F(x)$;

• 2.

($\Gamma-\limsup$) for all $x$ in $X$, there exists a sequence $\left\{x_{n}\right\}_{n}$ in $X$ such that $x_{n}\rightarrow x$ and $\limsup_{n\rightarrow+\infty}F_{n}\left(x_{n}\right)\leq F(x)$. Such a sequence is called a recovery sequence.

Lemma 2** ([33, Proposition 2.6]).**

Assume that $F_{n}$ $\Gamma$-converges to $F$. If for every $n$, $x_{n}$ minimizes $F_{n}$, and if the sequence $\left\{x_{n}\right\}_{n\in\mathbb{N}}$ converges to some $x$ in $X$, then $x$ minimizes $F$, and moreover, $\lim_{n\rightarrow+\infty}\min_{X}F_{n}=\min_{X}F$.

5.2. $\Gamma$-convergence of the $p$-adic Coulomb gas Hamiltonian

We denote by $\mathcal{P}(\mathbb{Q}_{p}^{d})$ the space of probability measures on $\mathbb{Q}_{p}^{d}$. By using the following map:

[TABLE]

which associates to the configuration of $n$ points the probability measure $\frac{1}{n}\sum\limits_{i=1}^{n}\delta_{x_{i}}$ (called the empirical measure), here $\delta_{x_{i}}$ denotes the Dirac distribution at $x_{i}$, we consider $H_{n}\left(x_{1},\ldots,x_{n}\right)$ as a function on $\mathcal{P}(\mathbb{Q}_{p}^{d})$, as follows:

[TABLE]

Theorem 1**.**

Assume that $d>\alpha$ and that $V$ is a continuous bounded from below function. The sequence $\left\{\frac{1}{n^{2}}H_{n}\right\}_{n}$ of functions (defined on $\mathcal{P}(\mathbb{Q}_{p}^{d})$) $\Gamma$-converges, with respect to the weak convergence of probability measures, to the function $I_{\alpha}:\mathcal{P}(\mathbb{Q}_{p}^{d})\rightarrow\left(-\infty,+\infty\right]$ defined by

[TABLE]

Remark 1**.**

From the point of view of statistical mechanics, $I$ is the mean-field limit energy of $H_{n}$.

The proof of this result will be given in Section 7.1.

5.3. Minimizing the mean-field energy via potential theory

In this section we consider the following minimization problem:

[TABLE]

Lemma 3**.**

The functional $I$ is strictly convex on $\mathcal{P}(\mathbb{Q}_{p}^{d})$.

Proof.

Since $\mu\rightarrow\int Vd\mu$ is linear, it is sufficient to show that the functional (the mutual energy of the measures $\mu$, $\nu$)

[TABLE]

satisfies

[TABLE]

for $0<\lambda<1$ and $\mu$, $\nu$ belonging to the convex cone of probability measures, with $\mathcal{E}_{\alpha}(\mu,\mu)$, $\mathcal{E}_{\alpha}(\nu,\nu)<+\infty$. This fact follows from Proposition 2. ∎

As a consequence, if there exists a minimizer to (5.1), it is unique. This minimizer is called the equilibrium measure or the Frostman equilibrium measure in potential theory. In order to show the existence of an equilibrium measure we make the following assumptions on the potential $V$:

[TABLE]

The first condition assures the lower semi-continuity of $I$ and that $\inf I>-\infty$. The second condition is equivalent to the condition $\lim_{\left\|x\right\|_{p}\rightarrow+\infty}V(x)=+\infty$.

Lemma 4** ([33, Lemma 2.10]).**

Assume that (A1) and (A2) are satisfied. Let $\left\{\mu_{n}\right\}_{n}$ be a sequence in $\mathcal{P}(\mathbb{Q}_{p}^{d})$ such that $\left\{I\left(\mu_{n}\right)\right\}_{n}$ is bounded. Then, up to extraction of a subsequence, $\left\{\mu_{n}\right\}_{n}$ converges to some $\mu$ in $\mathcal{P}(\mathbb{Q}_{p}^{d})$ in the weak sense of probabilities and

[TABLE]

Definition 3**.**

We define the capacity of a compact set $K\subset\mathbb{Q}_{p}^{d}$ by

[TABLE]

where $\mathcal{P}(K)$ denotes the set of probability measures supported in $K$, and $\mathcal{E}_{\alpha}(\mu,\mu)$ denotes the Coulomb energy defined as in (5.2). The capacity of $K$ is $+\infty$ if there no exists a probability measure $\mu\in\mathcal{P}(K)$ such that $\mathcal{E}_{\alpha}(\mu,\mu)<+\infty$. For a general set $E\subset\mathbb{Q}_{p}^{d}$, we set

[TABLE]

where $K$ runs through all the compact subsets of $E$.

Alternatively, we can define the capacity of an arbitrary set $A\subset\mathbb{Q}_{p}^{d}$ as $\operatorname{Cap}_{\alpha}(A)=\frac{1}{\inf\mathcal{E}_{\alpha}(\mu,\mu)}$, where $\mu$ runs through all the positive measures concentrated on $A$ with total mass $\mu\left(\mathbb{Q}_{p}^{d}\right)=\mu\left(A\right)=1$. The result would be the same if the support of $\mu$ is required to be compact and contained in $A$, see e.g. [15, Lemma 2.2.2].

The capacity is an increasing function. In addition, it satisfies the following:

Lemma 5** ([15, Lemma 2.3.1]).**

Let $N$ be a subset of $\mathbb{Q}_{p}^{d}$. The following conditions are equivalent:

• (i)

$\operatorname{Cap}_{\alpha}(N)=0$;

• (ii)

$\mu=0$* is the only positive measure of finite energy (i.e. $\mathcal{E}_{\alpha}(\mu,\mu)<+\infty$) concentrated on $N$;*

• (iii)

$\mu=0$* is the only positive measure of finite energy supported by some compact subset of $N$.*

A property is said to hold quasi-everywhere (q.e.), if it holds everywhere except on a set of capacity zero.

In order to allow the potential $V$ to take the value $+\infty$, which is equivalent to impose the constraint that the probability measures are supported only on a specific set, the set where $V$ is finite, we impose on the potential $V$ the following condition:

[TABLE]

Lemma 6** ([33, Lemma 2.13]).**

Under the assumptions (A1)–(A3), we have

[TABLE]

Theorem 2**.**

Under the assumptions (A1)–(A3) and $d>\alpha$, the minimum of $I$ over $\mathcal{P}(\mathbb{Q}_{p}^{d})$ exists, is finite and is achieved by a unique $\mu_{0}$, which has compact support of positive capacity. In addition $\mu_{0}$ is uniquely characterized by the fact that

[TABLE]

where

[TABLE]

is the electrostatic potential generated by $\mu_{0}$, and

[TABLE]

Proof.

The proof of this theorem is a slight variation of the proof of Theorem 2.1 in Serfaty’s book [33]. The reason is that the argument given in [33]  works on a Polish space. More precisely, we need $\left(\mathbb{Q}_{p}^{d},\left\|\cdot\right\|_{p}\right)$ to be a complete countable metric space, in order to use Prokhorov’s theorem, see e.g. [5], and that a subset $K$ of $\mathbb{Q}_{p}^{d}$ is compact if and only if it is closed and bounded. We give just some comments about the proof. For the details the reader may consult [33].

Set $I:=\inf_{\mu\in\mathcal{P}(\mathbb{Q}_{p}^{d})}I(\mu)$. Then by Lemma 6, $\inf I<+\infty$, and there exists a sequence $\left\{\mu_{n}\right\}_{n}$ in $\mathcal{P}(\mathbb{Q}_{p}^{d})$ such that $I(\mu_{n})\rightarrow I$. Since the sequence $\left\{I(\mu_{n})\right\}_{n}$ is bounded, by Lemma 4, there exists a probability measure $\mu_{0}$ such that a subsequence of $\{\mu_{n}\}_{n\in\mathbb{N}}$ converges in probability to $\mu_{0}$ and

[TABLE]

Then, by the definition of $I$, $I(\mu_{0})=I$. The uniqueness of $\mu_{0}$ follows from Lemma 3. The proof now follows as in [33, Theorem 2.1]. ∎

6. The Coulomb gas confined into

the unit ball

We denote by $GL\left(\mathbb{Z}_{p},d\right)$ the group of all the matrices $\boldsymbol{g}$ of size $d\times d$ with entries in $\mathbb{Z}_{p}$ satisfying $\left|\det\boldsymbol{g}\right|_{p}=1$. This group preserves the norm $\left\|\cdot\right\|_{p}$, i.e.

[TABLE]

see e.g. [20, Lemma 3.16]. Given $\boldsymbol{g}\in GL\left(\mathbb{Z}_{p},d\right)$, we define the probability measure

[TABLE]

where $\mu_{0}$ is the equilibrium measure given in Theorem 2. Then $\mu_{\boldsymbol{g}}$ is a probability measure supported in $\boldsymbol{g}^{-1}K$, with $K=\operatorname{supp}\mu_{0}$. Furthermore,

[TABLE]

If the potential is a radial function, i.e. $V(x)=V(\left\|x\right\|_{p})$, then

[TABLE]

Now we set

[TABLE]

Then

[TABLE]

Consequently, the measure satisfies conditions (5.3), and by the uniqueness of $\mu_{0}$, we conclude that

[TABLE]

Which implies that

[TABLE]

i.e. the support $K$ of the measure $\mu_{0}$ is a union of spheres.

Proposition 3**.**

Consider the potential

[TABLE]

where $V_{0}$ is a positive real number. Then the equilibrium measure is the characteristic function of the unit ball, i.e. $\mu_{0}\left(x\right)=\Omega\left(\left\|x\right\|_{p}\right)$, and

[TABLE]

Proof.

The support of the equilibrium measure is contained in $\mathbb{Z}_{p}^{d}$. Notice that by the discussion presented at the beginning of this section we cannot conclude that the support of $\mu_{0}$ is the unit ball, see (6.1). So we proceed as follows. We compute a candidate to the equilibrium measure assuming that $\operatorname{supp}\mu_{0}=\mathbb{Z}_{p}^{d}$, then we verify that the proposed measure satisfies conditions (5.3).

By restricting the formula given in (5.3) to the unit ball, we have

[TABLE]

We apply the operator $\boldsymbol{D}^{\alpha}$, with domain $\left\{T\in\mathcal{D}^{\prime}:\left\|\xi\right\|_{p}^{\alpha}\widehat{T}\in\mathcal{D}^{\prime}\right\}$, to both sides in (6.4). We first notice that $g_{\alpha}\left(x\right)=\frac{1}{\left\|x\right\|_{p}^{d-\alpha}}$, $d>\alpha$, is locally integrable, and since $\mu_{0}\left(x\right)$ has compact support, then $\mu_{0}\ast g_{\alpha}\in\mathcal{D}^{\prime}$, and $\widehat{\mu_{0}\ast g_{\alpha}}\left(\xi\right)=\widehat{\mu_{0}}\left(\xi\right)\widehat{g_{\alpha}}\left(\xi\right)$. Furthermore, since the support of $\mu_{0}\left(x\right)$ by supposition is the unit ball, then

[TABLE]

where $\mathcal{T}_{y}$ is the translation operator defined as $\mathcal{T}_{y}\varphi\left(x\right)=\varphi\left(x-y\right)$, $\varphi\in\mathcal{D}$, and $\left(\mathcal{T}_{y}G,\varphi\right)=\left(G,\mathcal{T}_{-y}\varphi\right)$, $\varphi\in\mathcal{D}$, $G\in\mathcal{D}^{\prime}$, see e.g. [36, Chap. III, Proposition 3.17]. Then

[TABLE]

We now compute

[TABLE]

Then from (6.4)–(6.5), we obtain

[TABLE]

which implies that the distribution in the left side of (6.6) is supported in the unit ball. And since the product of distributions is associative,

[TABLE]

We now use that

[TABLE]

to obtain $\widehat{\mu_{0}}\left(\xi\right)=\frac{1-p^{-\alpha}}{1-p^{-d}}\left(C-\frac{V_{0}}{2}\right)\Omega\left(\left\|\xi\right\|_{p}\right)$, and hence

[TABLE]

Then necessarily $\frac{1-p^{-\alpha}}{1-p^{-d}}\left(C-\frac{V_{0}}{2}\right)=1$, which implies (6.3). Finally, the verification that $\mu_{0}\left(x\right)=\Omega\left(\left\|x\right\|_{p}\right)$ satisfies (5.3) is straightforward. ∎

7. Proof of the $\Gamma$-convergence and some consequences

7.1. Proof of Theorem 1

The proof is organized in the same form as the proof of Proposition 2.8 in [33]. In the proof the topology of $\mathbb{Q}_{p}^{d}$ comes into play, and consequently there are important differences with the classical case.

**Step 1. **($\Gamma-\liminf$) If $\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}\rightarrow\mu$, then

[TABLE]

For the proof of this assertion the reader may consult [33, pp. 23–24].

**Step 2. **($\Gamma-\limsup$) We have to construct a recovery sequence for each measure $\mu\in\mathcal{P}(\mathbb{Q}_{p}^{d})$ such that $I(\mu)<+\infty$. Similarly to the proof of Proposition 2.8 from [33] it is sufficient to prove the statement for compactly supported measures. Moreover, by considering the $\delta$-approximating sequence

[TABLE]

and the convolutions $\mu_{n}=\mu\ast\delta_{n}$ and repeating the corresponding part of the proof of Proposition 2.8 from [33], we may further assume that $\mu$ is supported in some ball $B_{L}^{d}=p^{-L}\mathbb{Z}_{p}^{d}$, $L\geq 0$, has a density in $\mathcal{D}_{\mathbb{R}}(\mathbb{Q}_{p}^{d})$, and that this density is bounded from below by $\epsilon>0$, from above by $p^{Kd}-1$ for some $K\in\mathbb{N}$ and its index of local constancy $l\left(\mu\right)$ satisfies $l\left(\mu\right)\geq-M_{0}$ for some $M_{0}\in\mathbb{N}$.

Step 3. Let us fix some $M\geq M_{0}$. There are $p^{(L+M)d}$ balls of radius $p^{-M}$ in the support of the measure $\mu$. Let us denote them as $B_{k}$, $1\leq k\leq p^{(L+M)d}$. In each of these balls consider $p^{(M+K)d}$ smaller balls of radius $p^{-2M-K}$.

Now we distribute $p^{2Md}$ points into these larger balls as follows. In each ball $B_{k}$ we place $\left[p^{2Md}\mu(B_{k})\right]+\epsilon_{k}$ points, here $[x]$ denotes the largest integer not exceeding $x$, and we take $\epsilon_{k}$ equal to [math] or $1$ so that the total number of distributed points equals $p^{2Md}$. The total number of points in the ball $B_{k}$ does not exceed

[TABLE]

that is there are sufficiently many smaller balls (of radius $p^{-2M-K}$) to choose at most one point in each smaller ball. In such way we may select $p^{2Md}$ points $x_{1},\ldots,x_{p^{2Md}}$ and the distance between any two points will be at least $p^{-2M-K+1}$.

Consider the measure $\mu_{M}:=p^{-2Md}\sum_{i=1}^{p^{2Md}}\delta_{x_{i}}$ and let us show that $\mu_{M}\rightharpoonup\mu$, $M\to+\infty$ in the weak sense of probabilities.

Let us fix a test function $\varphi\in\mathcal{D}_{\mathbb{R}}(\mathbb{Q}_{p}^{d})$ and let $M\geq M_{0}$ be such that the index of local constancy $l\left(\varphi\right)$ satisfies $l\left(\varphi\right)\geq-M$. Both the density of the measure $\mu$ and the function $\varphi$ are constant on the balls $B_{k}$ of radius $p^{-M}$. Denote by $b_{k}$ a point in the ball $B_{k}$. Then

[TABLE]

where $\epsilon_{k}$ denotes 0 or 1 depending on the selection of the points $x_{i}$. The last expression converges to zero as $M\to+\infty$ since

[TABLE]

Step 4. Let $M$ be fixed and the points $x_{1},\ldots,x_{p^{2Md}}$ be chosen as in Step 3. We denote by $\Delta$ the diagonal of $\mathbb{Q}_{p}^{d}\times\mathbb{Q}_{p}^{d}$, and set $\Delta^{c}:=\mathbb{Q}_{p}^{d}\times\mathbb{Q}_{p}^{d}\smallsetminus\Delta$. Then

[TABLE]

Consider

[TABLE]

where \Omega\bigl{(}\left\|x\right\|_{p}\bigr{)} is the characteristic function of $\mathbb{Z}_{p}^{d}$. The function

[TABLE]

is a test function supported in the ball $B_{L}^{d}$, where $\mu_{M}$ and $\mu$ are supported. By using that $\mu_{M}\rightharpoonup\mu$, we conclude that $E_{M}^{\left(1\right)}$ converges to

[TABLE]

Claim:

[TABLE]

Since \left(\Omega g_{\alpha}\right)\bigl{(}\left\|x\right\|_{p}\bigr{)}=\frac{\Omega\left(\left\|x\right\|_{p}\right)}{\left\|x\right\|_{p}^{d-\alpha}}\in L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p}^{d},dx\right) and $\mathcal{D}_{\mathbb{R}}(\mathbb{Z}_{p}^{d})$ is dense in $L_{\mathbb{R}}^{1}\left(\mathbb{Z}_{p}^{d},dx\right)$, given $\epsilon>0$ there exists $\varphi\in\mathcal{D}_{\mathbb{R}}(\mathbb{Z}_{p}^{d})$ satisfying

[TABLE]

Now, by using (7.1) and the Young’s inequality $\left\|\nu\ast f\right\|_{1}\leq\left\|f\right\|_{1}\left\|\nu\right\|,$ where $\nu$ is a finite Borel measure, $\|\nu\|$ is its total variation and $f\in L_{\mathbb{R}}^{1}$, we obtain

[TABLE]

hence

[TABLE]

which implies the announced Claim.

Therefore,

[TABLE]

(actually, equality holds).

On the other hand, since $V$ is continuous, $\mu_{M}\rightharpoonup\mu$, and $\mu_{M}$, $\mu$ are supported in $B_{L}^{d}$, we have $\int Vd\mu_{M}\rightarrow\int Vd\mu$.

In conclusion,

[TABLE]

7.2. Some further results

Lemma 7** ([33, Lemma 2.21]).**

Assume that $V$ satisfies (A1)–(A2). Let

[TABLE]

be a sequence of configurations, and let $\left\{\mu_{n}\right\}_{n}$ be associated empirical measures (defined by $\mu_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}$). Assume that $\left\{\frac{1}{n^{2}}H_{n}\left(x_{1},\ldots,x_{n}\right)\right\}_{n}$ is a bounded sequence. Then the sequence $\left\{\mu_{n}\right\}_{n}$ is tight, and as $n\rightarrow+\infty$, it converges weakly in $\mathcal{P}(\mathbb{Q}_{p}^{d})$ (up to extraction of a subsequence) to some probability measure $\mu$.

Theorem 3**.**

Assume that $V$ is continuous and satisfies (A2). Assume that for each $n$, $\left\{\left(x_{1},\ldots,x_{n}\right)\right\}_{n}$ is a minimizer of $H_{n}$. Then

[TABLE]

where $\mu_{0}$ is the unique minimizer of $I$ as in Theorem 2, and

[TABLE]

Proof.

The proof follows from Lemma 7, Proposition 2 and Theorems 1, 2, by using the reasoning given in [33] for the proof of Theorem 2.2. ∎

7.3. Continuum limits of hierarchical models

The energy function $-I\left(\mu\right)$ is the continuum limit of a $p$-adic hierarchical Hamiltonian, which corresponds to a certain type of $p$-adic hierarchical spin glass. A similar result was established by Lerner and Missarov [21, Theorem 2], see also [16, Section C]. For $L\in\mathbb{Z}$ fixed, we take the potential

[TABLE]

where $V_{0}:B_{L}^{d}\rightarrow\mathbb{R}$ is a continuous function. Let $\mathcal{P}(B_{L}^{d})$ denote the space of probability distributions supported in $B_{L}^{d}$. Consider the functional

[TABLE]

for $\mu\in\mathcal{P}(B_{L}^{d})$. There exists a probability measure $\mu_{0}\in\mathcal{P}(B_{L}^{d})$ such that $\min_{\mu\in\mathcal{P}(B_{L}^{d})}I_{L}(\mu)=I_{L}(\mu_{0})$.  This fact follows from Theorem 2 by noticing that the equilibrium measure must be supported in $B_{L}^{d}$ due to the fact that the potential $V$ is infinite outside of this ball.

We set $G_{l}:=p^{-L}\mathbb{Z}_{p}^{d}/p^{l}\mathbb{Z}_{p}^{d}$, with $l\geq-L$.  By fixing an identification of $G_{l}$ with a subset of $\mathbb{Q}_{p}^{d}$, $G_{l}$ becomes a finite ultrametric space, see e.g. [39, Section 3]. We also pick $\rho:B_{L}^{d}\rightarrow\left[0,\infty\right)$ a continuous function. We now define the following approximations of $\rho$ and $V_{0}$:

[TABLE]

and

[TABLE]

which are test functions supported in $B_{L}^{d}$ satisfying that $\rho_{l}\overset{\left\|\cdot\right\|_{\infty}}{\rightarrow}\rho$ and $V_{l}\overset{\left\|\cdot\right\|_{\infty}}{\rightarrow}V_{0}$, see e.g. [39, Lemma 1]. Then

[TABLE]

where

[TABLE]

The function $-I_{L}(\rho_{l}\,dx)$ is the Hamiltonian of a spin glass with $p$-adic coupling, see [16, Section C]. We now show that

[TABLE]

Indeed, since $\rho_{l}\overset{\left\|\cdot\right\|_{\infty}}{\rightarrow}\rho$, $V_{l}\overset{\left\|\cdot\right\|_{\infty}}{\rightarrow}V_{0}$, there is a positive constant $C$ such that $\left\|\rho_{l}\right\|_{\infty}<C\left\|\rho\right\|_{\infty}$ and $\left\|V_{l}\right\|_{\infty}<C\left\|V_{0}\right\|_{\infty}$ for $l$ sufficiently large. Consequently by the dominated convergence lemma,

[TABLE]

and

[TABLE]