Coulomb Random Point Field Overview

• Coulomb random point fields are random point processes defined by Coulomb interactions that model Gibbs ensembles and infinite-particle equilibria.
• They use renormalized energy functionals to capture microscopic fluctuations, underpinning Gaussian free field limits and crystallization effects.
• Stochastic dynamics formulations recast these fields as equilibrium measures, enabling insights into long-range interactions and hyperuniformity.

Searching arXiv for recent and foundational papers on Coulomb random point fields. A Coulomb random point field is, in the most common usage of the recent literature, a random point process generated by Coulomb-interacting particles, either as the microscopic or thermodynamic limit of a Gibbs ensemble, or as the infinite-particle equilibrium measure on configuration space for Coulomb interacting Brownian motions. The term does not denote a single universally fixed formal object across the literature: some reviews explicitly treat it as a broad family of Coulomb-interacting point processes, while stochastic-dynamics papers use it for the reversible infinite-volume equilibrium measure itself (Serfaty, 2017, Ghosh et al., 2018, Osada et al., 29 Aug 2025). Canonical examples include the Ginibre random point field in two dimensions, Dyson-type limits in one dimension, microscopic empirical fields selected by renormalized energy plus entropy, and a range of determinantal or Gibbsian limits arising from weak confinement, boundary scaling, and large-deviation conditioning (Osada, 2010).

1. Finite-particle Coulomb systems and macroscopic equilibrium

The basic finite-particle model is a configuration of $N$ points $x_1,\dots,x_N\in\mathbb R^d$ with energy

$$\mathcal H_N(x_1,\dots,x_N)=\sum_{i\neq j} g(x_i-x_j)+N\sum_{i=1}^N V(x_i),$$

where $g$ is the Coulomb kernel and $V$ is an external confining potential. In dimension $d=2$, $g(x)=-\log|x|$; in dimensions $d\ge 3$, $g(x)=|x|^{2-d}$; the literature also treats the one-dimensional logarithmic case and more general Riesz kernels $g(x)=|x|^{-s}$ (Serfaty, 2017).

The associated Gibbs measure at inverse temperature $x_1,\dots,x_N\in\mathbb R^d$0 is

$x_1,\dots,x_N\in\mathbb R^d$1

while a standard two-dimensional normalization is

$x_1,\dots,x_N\in\mathbb R^d$2

with

$x_1,\dots,x_N\in\mathbb R^d$3

These formulations encode the same structural ingredients: pairwise Coulomb repulsion and confinement (Leblé et al., 2016).

The macroscopic observable is the empirical measure

$x_1,\dots,x_N\in\mathbb R^d$4

Its large-$x_1,\dots,x_N\in\mathbb R^d$5 behavior is governed by the mean-field functional

$x_1,\dots,x_N\in\mathbb R^d$6

whose minimizer $x_1,\dots,x_N\in\mathbb R^d$7 is the equilibrium measure. In the Coulomb setting it is characterized by the Euler–Lagrange condition

$x_1,\dots,x_N\in\mathbb R^d$8

and minimizers of $x_1,\dots,x_N\in\mathbb R^d$9 satisfy $$\mathcal H_N(x_1,\dots,x_N)=\sum_{i\neq j} g(x_i-x_j)+N\sum_{i=1}^N V(x_i),$$0 (Serfaty, 2017).

A closely related deterministic viewpoint appears in mean-field convergence for Coulomb-type flows. There the modulated energy

$$\mathcal H_N(x_1,\dots,x_N)=\sum_{i\neq j} g(x_i-x_j)+N\sum_{i=1}^N V(x_i),$$1

acts as a Coulomb or Riesz distance between the discrete configuration and the limiting continuum density, and yields quantitative propagation of the mean-field limit in arbitrary dimension (Serfaty et al., 2018).

2. Microscopic point fields, renormalized energy, and free energy

Beyond the macroscopic law of large numbers, Coulomb random point field theory is concerned with the blown-up microscopic configuration at scale $$\mathcal H_N(x_1,\dots,x_N)=\sum_{i\neq j} g(x_i-x_j)+N\sum_{i=1}^N V(x_i),$$2. In the review literature, the relevant objects are tagged empirical fields and point processes on local configuration space, rather than only empirical measures. The microscopic energy is described by a renormalized electric-energy functional $$\mathcal H_N(x_1,\dots,x_N)=\sum_{i\neq j} g(x_i-x_j)+N\sum_{i=1}^N V(x_i),$$3, defined for an electric field $$\mathcal H_N(x_1,\dots,x_N)=\sum_{i\neq j} g(x_i-x_j)+N\sum_{i=1}^N V(x_i),$$4 solving

$$\mathcal H_N(x_1,\dots,x_N)=\sum_{i\neq j} g(x_i-x_j)+N\sum_{i=1}^N V(x_i),$$5

where $$\mathcal H_N(x_1,\dots,x_N)=\sum_{i\neq j} g(x_i-x_j)+N\sum_{i=1}^N V(x_i),$$6 is an infinite point configuration of unit density against a neutralizing background (Serfaty, 2017).

The microscopic Gibbs principle is then formulated as an energy-entropy variational problem. One version is

$$\mathcal H_N(x_1,\dots,x_N)=\sum_{i\neq j} g(x_i-x_j)+N\sum_{i=1}^N V(x_i),$$7

where $$\mathcal H_N(x_1,\dots,x_N)=\sum_{i\neq j} g(x_i-x_j)+N\sum_{i=1}^N V(x_i),$$8 is a microscopic point process. Another, process-level formulation is

$$\mathcal H_N(x_1,\dots,x_N)=\sum_{i\neq j} g(x_i-x_j)+N\sum_{i=1}^N V(x_i),$$9

with $g$0 the unit-intensity Poisson process (Serfaty, 2017, Leblé, 2015). The first term favors low electrostatic energy and ultimately crystalline order; the second favors disorder.

A central contribution of the process-level theory is a unified definition of logarithmic, Coulomb, and Riesz energy for infinite neutral point configurations with background charge. Besides the electric formulation, there is an intrinsic formulation in terms of the two-point function $g$1: $g$2 The theory shows that, in the settings where equivalence is proved, the electric renormalized energy is the lower semicontinuous regularization of the intrinsic correlation-function energy, possibly with the logarithmic discrepancy correction $g$3 in log-gases (Leblé, 2015).

This framework yields two extremal regimes. As $g$4, minimizers of the free energy converge to the Poisson point process: $g$5 In the one-dimensional logarithmic and Riesz cases, as $g$6, minimizers converge to the shifted lattice process

$g$7

which is the crystallization limit (Leblé, 2015). This suggests a general interpretation of a Coulomb random point field as the microscopic equilibrium process selected by the competition of renormalized energy and specific entropy.

3. Fluctuations, Gaussian fields, and hyperuniformity

For two-dimensional Coulomb gases, fluctuation theory identifies the point field with a nontrivial Gaussian field at large scales. A fundamental result proves a Central Limit Theorem for linear statistics at arbitrary inverse temperature, for general confining potential, at macroscopic and mesoscopic scales, and possibly near the boundary of the support of the equilibrium measure. In the interior mesoscopic regime, if

$g$8

then

$g$9

and the full result is interpreted as convergence of the random electrostatic potential to a Gaussian Free Field (Leblé et al., 2016).

This fluctuation theory is one rigorous expression of rigidity: linear statistics fluctuate on scales far below the i.i.d. benchmark. The same work also proves moderate deviation upper bounds

$V$0

and identifies the stronger zero-temperature behavior of energy minimizers (Leblé et al., 2016).

In higher-dimensional Euclidean Coulomb gases, comparable fluctuation statements remain harder. Hierarchical models provide a tractable surrogate. In the three-dimensional hierarchical Coulomb gas, for an open set $V$1 with regular boundary,

$V$2

while the fluctuations are also proved to be at least of order $V$3, matching the classical cube-root prediction up to logarithmic loss. After rescaling to microscopic scale, the variance in a bounded window is only logarithmic in the window diameter (Chatterjee, 2017).

The all-dimensional hierarchical theory strengthens this picture. For smooth connected open $V$4, the number variance has lower and upper bounds of surface-area type up to logarithmic factors, dyadic cubes satisfy

$V$5

and Lipschitz linear statistics obey

$V$6

The ground states are characterized recursively by the condition that in every dyadic cube the child occupancies differ by at most $V$7, which is the hierarchical analogue of crystallization (Ganguly et al., 2019).

A complementary diagnostic is the spatial form factor

$V$8

introduced as the averaged even Fourier transform of pair distances. In Coulomb gases it provides a one-dimensional probe of interaction-induced departures from the Poisson baseline, separating geometric from genuinely interaction-driven effects (Massaro et al., 2024).

4. Geometry, topology, boundaries, and rare events

Weak confinement produces a qualitatively different Coulomb random point field: a macroscopic outlier process outside the charged support. For a determinantal planar Coulomb gas at $V$9, with an uncharged region

$d=2$0

the outlier process is

$d=2$1

Its limit depends only on the geometry of $d=2$2 and the global excess charge. In a simply connected hole and for $d=2$3,

$d=2$4

the Bergman point process on $d=2$5. For finitely connected $d=2$6, the subsequential limits form a family of weighted Bergman point processes indexed by

$d=2$7

and outliers in distinct uncharged components are asymptotically independent, even when the components share boundary points. The paper interprets this as screening (Butez et al., 2021).

Boundary scaling yields another family of limiting point fields. In planar determinantal Coulomb gases for random normal matrices, the interpolation

$d=2$8

connects free boundary ($d=2$9), hard edge ($g(x)=-\log|x|$0), and weaker-than-free or fuzzy boundaries ($g(x)=-\log|x|$1). The rescaled edge process has a universal determinantal limit with intensity

$g(x)=-\log|x|$2

At the ultraweak endpoint $g(x)=-\log|x|$3, the theory produces a new point field with heavy-tail behavior

$g(x)=-\log|x|$4

and this is explicitly described as a new Coulomb random point field (Ameur et al., 2019).

Rare-event geometry is organized by large deviations. For the Ginibre ensemble, the hole probability in a disk satisfies

$g(x)=-\log|x|$5

whereas for Gaussian entire function zeros

$g(x)=-\log|x|$6

More generally, conditioned configurations concentrate near constrained minimizers of the rate functional; in the Ginibre case these can be approximated by weighted Fekete points, while in Gaussian entire function zeros the conditioned limit exhibits a forbidden annulus outside the hole (Ghosh et al., 2018).

A recent spherical development studies two-dimensional Coulomb gases on the Riemann sphere with determinantal or Pfaffian structures, under external potentials invariant under rotations around the axis connecting the north and south poles, and with microscopic point charges inserted at the poles. These models are interpreted as weakly confining planar Coulomb gases whose droplet is the entire complex plane, and the work derives precise asymptotic expansions of the free energies, including the constant terms (Byun et al., 13 Jan 2025).

5. Infinite-particle equilibrium measures and stochastic dynamics

In stochastic-dynamics papers, a Coulomb random point field is the equilibrium measure of an infinite Brownian system with Coulomb interaction. The configuration space is

$g(x)=-\log|x|$7

or equivalently the locally finite configuration space on $g(x)=-\log|x|$8 (Osada, 2010, Osada et al., 29 Aug 2025).

For the two-dimensional Coulomb interaction

$g(x)=-\log|x|$9

the formal infinite-dimensional SDE is

$d\ge 3$0

A foundational result solves such ISDEs when the equilibrium state is the Ginibre random point field or Dyson’s measures, using an integration by parts formula and the logarithmic derivative of the equilibrium measure. In the Ginibre case,

$d\ge 3$1

in $d\ge 3$2 for $d\ge 3$3. The same diffusion also satisfies an alternative “plural ISDE,” a phenomenon emphasized as specific to the Ginibre field (Osada, 2010).

A central conceptual point is that the two-dimensional Coulomb interaction is not a Ruelle-class potential. Standard Gibbs/DLR machinery does not apply in the usual way, the drift is not absolutely convergent, and the long-range tail qualitatively changes the dynamics (Osada, 2010). This addresses a common misconception: Coulomb random point fields are not merely ordinary Gibbs point processes with a singular pair potential.

The recent all-dimensional theory extends this program to arbitrary $d\ge 3$4 and all $d\ge 3$5. For

$d\ge 3$6

the Coulomb random point field is defined as the infinite-particle equilibrium measure obtained as a weak limit of finite-particle measures. Its logarithmic derivative is explicitly

$d\ge 3$7

and the ISDE

$d\ge 3$8

admits weak solutions under general assumptions and pathwise unique strong solutions under stronger ones. The unlabeled process is reversible with respect to the Coulomb random point field, whereas the labeled diffusion has no invariant measure of the same type; finite-particle systems converge to the infinite dynamics (Osada et al., 29 Aug 2025).

6. Conceptual scope and terminology

The literature supports two closely related, but not identical, uses of the term. In the review tradition, the closest interpretation is “a random point process arising as the microscopic or thermodynamic limit of a Gibbs ensemble of Coulomb-interacting particles” (Serfaty, 2017). In the stochastic-dynamics tradition, it is “the infinite-particle equilibrium measure on configuration space for particles in $d\ge 3$9 interacting through the $g(x)=|x|^{2-d}$0-dimensional Coulomb potential,” with respect to which the unlabeled diffusion is reversible (Osada et al., 29 Aug 2025).

This terminological plurality is not accidental. The subject spans several mathematically distinct regimes: finite-$g(x)=|x|^{2-d}$1 Gibbs ensembles; equilibrium measures and obstacle problems; renormalized microscopic fields; determinantal special cases such as Ginibre, Bergman, and random-normal-matrix edge fields; large-deviation conditioned processes; and infinite-dimensional diffusions (Leblé et al., 2016, Butez et al., 2021). The common structure is the same: long-range Coulomb repulsion, a neutralizing background or confining mechanism, and a point-process description of local particle statistics.

A plausible implication is that “Coulomb random point field” functions less as the name of a single model than as a unifying category for equilibrium and near-equilibrium point processes generated by Coulomb interaction. Within that category, several recurrent phenomena are rigorously established: convergence of linear statistics to a Gaussian Free Field, screening and asymptotic independence across separated holes, hyperuniform or surface-area-order number fluctuations in tractable models, crystallization tendencies in low-temperature limits, and reversible infinite-particle stochastic dynamics (Leblé et al., 2016, Ganguly et al., 2019, Osada et al., 29 Aug 2025).