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).
This article extends recent results on log-Coulomb gases in a $p$-field $K$ (i.e., a nonarchimedean local field) to those in its projective line $\mathbb{P}^1(K)$, where the latter is endowed with the $PGL_2$-invariant Borel probability measure and spherical metric. Our first main result is an explicit combinatorial formula for the canonical partition function of log-Coulomb gases in $\mathbb{P}^1(K)$ with arbitrary charge values. Our second main result is called the "$(q+1)$th Power Law", which relates the grand canonical partition functions for one-component gases in $\mathbb{P}^1(K)$ (where…
These are the lecture notes of a course taught at the Park City Mathematics Institute in June 2017. They are intended to review some recent results, obtained in large part with Thomas Leblé, on the statistical mechanics of systems of points with logarithmic or Coulomb interactions. After listing some motivations, we describe the "electric approach" which allows to get concentration results, Central Limit Theorems for fluctuations, and a Large Deviations Principle expressed in terms of the microscopic state of the system.
We consider the Coulomb gas of $N$ particles on the sphere and show that the logarithmic energy of the configurations approaches the minimal energy up to an error of order $\log N$, with exponentially high probability and on average, provided the temperature is $\mathcal O(1/N)$.
The two dimensional Coulomb gas is the prototypical model of statistical mechanics displaying a special kind of phase transition, named after Berezinskii, Kosterlitz and Thouless. Physicists and mathematicians proposed several predictions about this system. Two of them, valid along the phase transition curve and for small activity, are: a) the long-distance decay of the "fractional charge" correlation is power law, with a multiplicative logarithmic correction; b) in such a decay, the exponent of the power law, as well as the exponent of the logarithmic correction, have a certain precise depend…
A random set of points in Euclidean space is called `rigid' or `hyperuniform' if the number of points falling inside any given region has significantly smaller fluctuations than the corresponding number for a set of i.i.d. random points. This phenomenon has received considerable attention in recent years, due to its appearance in random matrix theory, the theory of Coulomb gases and zeros of random analytic functions. However, most of the published results are in dimensions one and two. This paper gives the first proof of hyperuniformity in a Coulomb type system in dimension three, known as th…
We study a log-gas on a network (a finite, simple graph) confined in a bounded subset of a local field (i.e. R, C, Q_{p} the field of p-adic numbers). In this gas, a log-Coulomb interaction between two charged particles occurs only when the sites of the particles are connected by an edge of the network. The partition functions of such gases turn out to be a particular class of multivariate local zeta functions attached to the network and a positive test function which is determined by the confining potential. The methods and results of the theory of local zeta functions allow us to establish t…
We examine a Coulomb gas consisting of $n$ identical repelling point charges at an arbitrary inverse temperature $β$, subjected to a suitable external field. We prove that the gas is effectively localized to a small neighbourhood of the "droplet" -- the support of the equilibrium measure determined by the external field. More precisely, we prove that the distance between the droplet and the vacuum is with very high probability at most proportional to $$\sqrt{\dfrac {\log n}{βn}}.$$ This order of magnitude is known to be "tight" when $β=1$ and the external field is radially symmetric. In additi…
We consider planar Coulomb systems consisting of a large number $n$ of repelling point charges in the low temperature regime, where the inverse temperature $β$ grows at least logarithmically in $n$ as $n \longrightarrow \infty$, i.e., $β\gtrsim \log n$. Under suitable conditions on an external potential we prove results to the effect that the gas is with high probability uniformly separated and equidistributed with respect to the corresponding equilibrium measure (in the given external field). Our results generalize earlier results about Fekete configurations, i.e., the case $β=\infty$. There …
We consider a one-dimensional classical Coulomb gas of $N$ like-charges in a harmonic potential -- also known as the one-dimensional one-component plasma (1dOCP). We compute analytically the probability distribution of the position $x_{\max}$ of the rightmost charge in the limit of large $N$. We show that the typical fluctuations of $x_{\max}$ around its mean are described by a non-trivial scaling function, with asymmetric tails. This distribution is different from the Tracy-Widom distribution of $x_{\max}$ for the Dyson's log-gas. We also compute the large deviation functions of $x_{\max}$ ex…
We consider the normal matrix ensemble under a general confining potential. We find that the eigenvalues condensate on a compact set in the plane, which we call the spectral droplet. We also study the evolution of incrementally adding a dimension, i.e., adding an extra electron in this fermionic model.
We investigate a Coulomb gas in a potential satisfying a weaker growth assumption than usual and establish a large deviation principle for its empirical measure. As a consequence the empirical measure is seen to converge towards a non-random limiting measure, characterized by a variational principle from logarithmic potential theory, which may not have compact support. The proof of the large deviation upper bound is based on a compactification procedure which may be of help for further large deviation principles.
We have developed a field theory for strongly coupled Coulomb fluids, via introducing new functional--integral transformation of the electrostatic interaction energy. Our formalism not only reproduces the Lieb--Narnhofer lower bound, but also bridges logical gaps which previous approaches have involved.
We study the long-time behavior of the dynamics of interacting planar Brow-nian particles, confined by an external field and subject to a singular pair repulsion. The invariant law is an exchangeable Boltzmann -- Gibbs measure. For a special inverse temperature, it matches the Coulomb gas known as the complex Ginibre ensemble. The difficulty comes from the interaction which is not convex, in contrast with the case of one-dimensional log-gases associated with the Dyson Brownian Motion. Despite the fact that the invariant law is neither product nor log-concave, we show that the system is well-po…
We establish sharp deviation inequalities for the linear statistics of the 2D Coulomb gas. These imply sub-Gaussian inequalities, where the variance is given by the Dirichlet norm. The proofs use complex geometry and potential theory on Riemann surfaces and apply more generally to beta-ensembles, which also include integer Quantum Hall states on Riemann surfaces. In a sequel of the paper we give applications to large and moderate deviation principles, local laws at mesoscopic scales, quantitative Bergman kernel asymptotics. In a series of companion papers applications to concentration of measu…
We consider determinantal Coulomb gas ensembles with a class of discrete rotational symmetric potentials whose droplets consist of several disconnected components. Under the insertion of a point charge at the origin, we derive the asymptotic behaviour of the correlation kernels both in the macro- and microscopic scales. In the macroscopic scale, this particularly shows that there are strong correlations among the particles on the boundary of the droplets. In the microscopic scale, this establishes the edge universality. For the proofs, we use the nonlinear steepest descent method on the matrix…
We consider the generating functional (logarithm of the normalization factor) of the Laughlin state on a sphere, in the limit of a large number of particles $N$. The problem is reformulated in terms of a perturbative expansion of a 2d QFT, resembling the Liouville field theory. We develop an analog of the Liouville loop perturbation theory, which allows us to quantitatively study the generating functional for an arbitrary smooth metric and an inhomogeneous magnetic field beyond the leading orders in large $N$.
