Super-Coulombic Riesz Gas Overview

• Super-Coulombic Riesz gas is defined by particles in ℝ^d interacting via a singular Riesz potential |x-y|⁻ˢ with s>d-2, leading to strong short-range repulsion.
• Key methodologies include sharp scaling limits, energy splitting into macroscopic and microscopic components, and explicit mean-field convergence rates underpinning crystallization phenomena.
• Recent advances unify microscopic, mesoscopic, and macroscopic analyses to reveal robust local laws, a mesoscopic CLT, and comprehensive fluctuation theory.

A super-Coulombic Riesz gas is a system of $N$ interacting particles in $\mathbb{R}^d$ with pairwise repulsion governed by the Riesz kernel $|x-y|^{-s}$, where the exponent $s$ lies strictly above the Coulomb threshold: $s > d-2$. This regime is marked by strongly singular, effectively short-range repulsions and supports rich thermodynamic, kinetic, and statistical phenomena ranging from equilibrium crystallization to universality in fluctuation theory and explicit mean-field convergence rates. Recent works have unified microscopic, mesoscopic, and macroscopic perspectives on such gases, introducing robust analytic structures, sharp scaling limits, and localization properties.

1. Definition and Canonical Models

The super-Coulombic Riesz gas generalizes classical Coulomb and log-gas systems by enforcing a repulsive potential $W_s(x) = |x|^{-s}$, $s > d-2$, for configurations $(x_1, \dots, x_N) \in \mathbb{R}^{dN}$ under confining external potentials $V$. The Hamiltonian in its canonical form is:

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

with Gibbs law at inverse temperature $\beta$:

$$P_{N,\beta}(dx_1 \dots dx_N) = Z_{N,\beta}^{-1} \exp(-\beta H_N(x_1, \dots, x_N)) dx_1 \dots dx_N$$

The interaction is strictly more singular than Coulomb $s = d-2$ and log-gas $s=0$ cases, rendering kinetic constraints and equilibrium structures distinct.

Effective Hydrodynamics and Single-File Universality

In dimension $d=1$, the overdamped dynamics under Riesz interaction exhibit single-file behavior for $s > 1$, where the dynamics are dominated by hard-core repulsion resulting in equidistant local configurations. The coarse-grained free energy becomes local, and deterministic current $J_{\rm det}$ satisfies Fick's law with density-dependent diffusion:

$$J_{\rm det} = -D(\rho)\partial_x q, \qquad D(\rho) = D + (1+s)\zeta(s)g\rho^s$$

The variance of tracer displacement and integrated current both scale sub-diffusively, with standard deviations proportional to $t^{1/4}$ and amplitudes $A(s), B(s)$ determined explicitly by microscopic parameters (Dandekar et al., 2022).

2. Equilibrium Structure, Next-Order Asymptotics, and Renormalized Energy

Macroscopic and Microscopic Energy Splitting

Under weak confinement, the equilibrium measure $\mu_V$ minimizes the functional

$$I_s^V(\mu) = \iint |x - y|^{-s} d\mu(x)d\mu(y) + \int V(x)d\mu(x)$$

with compact support $\Sigma = \mathrm{supp}(\mu_V)$. The minimal energy admits a two-scale expansion (Cotar et al., 2017, Petrache et al., 2016, Leblé, 2015):

$$E_N(s) = N^2 I_s^V(\mu_V) + N^{1+s/d} C_*\int_{\mathbb{R}^d} \mu_V^{1+s/d} dx + o(N^{1+s/d})$$

where $C_* = C_{\mathrm{Jel}} = C_{\mathrm{UEG}}$ (upon crystallization), and $N^{1+s/d}$ is the next-order term universally controlled by the jellium (renormalized) energy.

Renormalized (Jellium) Energy and Screening

Microscopically, the arrangement of particles is governed by the minimization of the renormalized energy $\mathcal{W}$, defined via Caffarelli–Silvestre extension for $s > d-2$:

$$\mathcal{W}(E) = \lim_{R\to\infty} \frac{1}{R^d} \int_{K_R\times\mathbb{R}} |y|^\gamma |\nabla h|^2 - c_{s,d}g(\eta) \#\{\text{points}\}$$

and similarly for random point processes, the infinite-volume free energy combines $\mathcal{W}$ with entropy (Leblé, 2015)

$$\mathcal{F}_\beta(P) = W_s^{\rm ren}(P) + \frac{1}{\beta} H(P|\Pi)$$

where $P$ is a stationary point process, $\Pi$ is the Poisson process.

Anisotropic Interactions

For general anisotropic Riesz kernels $W_s(x) = |x|^{-s} \Phi(x/|x|)$ with physical ellipsoidal confinement, the equilibrium measure is independent of anisotropy in the super-Coulombic regime. The minimizer is the push-forward of the isotropic ball-law, yielding explicit densities and energies (Mora et al., 10 Jul 2025).

3. Fluctuation Theory, Local Laws, and Mesoscopic CLT

Local Laws and Screening

By bootstrapping energy and particle-number controls down to mesoscopic and microscopic scales, one obtains sharp local laws: in any bulk cube of side $\ell \gg N^{-1/d}$, both the localized next-order energy and the number of particles are controlled up to exponentially small probabilities (Peilen et al., 23 Nov 2025):

$P_{N,\beta}(G_\ell^c) \leq C e^{-c\beta \ell^d N}$

$$\tilde{F}_N^{\Box_\ell}(X_N) \leq C_\beta \ell^d N^{1+s/d}$$

This results in empirical process convergence and the existence of microscopic limit point processes.

Mesoscopic Central Limit Theorem

For linear statistics $L_N(f)$ on scales $\ell \gg N^{-1/d}$, a mesoscopic CLT emerges: fluctuations converge to a fractional Gaussian field with covariance (for smooth $f$)

$$\mathrm{Var}(L_N(f)) \sim (2\pi)^{-d} \int_{\mathbb{R}^d} |\hat{f}(k)|^2 |k|^{-d+s} dk$$

i.e., fluctuations scale as $N^{1/2+s/(2d)}$ (Peilen et al., 23 Nov 2025). At zero or finite temperature, centering by the equilibrium measure yields Gaussian statistics with fractional scaling.

4. Transport, Commutator Inequalities, and Mean-Field Limits

Sharp Functional Inequalities

Recent works (Rosenzweig et al., 22 Jul 2024) provide sharp commutator estimates of all orders for the Riesz modulated energy, proving that transport derivatives are controlled by the modulated energy itself, up to universally sharp additive errors $O(N^{s/d-1})$. The main analytic tool is the recursive PDE structure of generalized commutators $κ^{(n),f}$ under the degenerate elliptic operator $\mathcal{L} = -\mathrm{div}(|z|^\gamma \nabla)$ arising in the Caffarelli–Silvestre extension.

Propagation of Chaos and Mean-Field Convergence

For gradient flows and Hamiltonian dynamics with super-Coulombic Riesz interaction, mean-field convergence is achieved at the optimal rate $N^{s/d-1}$ under appropriate regularity:

$F_N(t) \leq e^{C t} [F_N(0) + O(N^{s/d-1})]$

where $F_N$ is the modulated energy distance, and empirical measures $\mu_N^t \rightharpoonup \mu^t$ weak-* converge to the limiting PDE solution (Nguyen et al., 2021, Serfaty et al., 2018). This analysis applies for dissipative, conservative, and mixed stochastic flows.

5. Statistical Properties, Pair Correlations, and Linear Statistics

Pair Correlation Functions

For $d=1$, super-Coulombic cases $s > 1$ lack universal closed-form for smoothed pair correlations. The relation between density and potential is nonlinear, and the correlation function $g_2$ rapidly decays beyond interparticle spacing (Beenakker, 2022):

• The pair correlation length $\sim$ interparticle spacing.
• No universal power-law tail; model-specific features dominate.

Linear Statistic Fluctuations and Large Deviations

Analytic formulae for cumulants and large deviation rate functions of linear statistics $\mathcal{L}_N$ are derived for 1D trapped Riesz gases, revealing edge singularities and evaporation transitions unique to the super-Coulombic window $-2 < k < -1$ (Doussal et al., 8 Aug 2024). The variance and higher cumulants deviate from Coulombic scaling, carrying explicit modifications due to the singular nature of the interaction.

6. Thermodynamics, Phase Behavior, and Universality

Screening and Crystallization

In low-temperature regimes, minimizers of the super-Coulombic renormalized energy correspond to crystalline configurations (e.g., integer lattice in $d=1$), supporting strong rigidity (Leblé, 2015). At high temperature, processes converge to Poisson statistics. Partition-function expansions show precise two-scale separation:

$$\log Z_{N,\beta} = -\beta N^2 I_V - \beta N \min_P \mathcal{F}_\beta(P) + o(N)$$

Emergent Collective Phenomena and BKT Transition

In $d=2$, super-Coulombic plasmas exhibit a charge confinement-deconfinement transition of Berezinskii-Kosterlitz-Thouless type, where dielectric screening induces a crossover from microscopically super-Coulombic interaction to an emergent Coulomb potential at large scales. The transition occurs universally across models and densities (De et al., 5 Jun 2025).

7. Open Problems, Conjectures, and Extensions

• The decay hypothesis for the Caffarelli–Silvestre extended field is proven only for periodic configurations; general proof for all minimizers remains an open problem (Petrache et al., 2016).
• Crystallization conjecture (periodic minimizer of $\mathcal{W}$) is established in $d=1$; extension to higher dimensions for arbitrary $s > d-2$ is an active research area.
• Extensions to anisotropic, nonlocal, and singular kernels—along with analysis on compact manifolds and quasi-neutral limits—are underway (Mora et al., 10 Jul 2025, Rosenzweig et al., 22 Jul 2024).
• Universality of fluctuations and additivity: recent mesoscopic CLTs suggest that a fractional Gaussian field arises as the universal limit for local statistics (Peilen et al., 23 Nov 2025).

---

In summary, the super-Coulombic Riesz gas regime ($s > d-2$) is typified by strong local singularity, effective short-range repulsion, and universality in both energy splitting and fluctuation law, with mean-field dynamics controlled by sharp functional inequalities, local laws, and robust screening. Crystallization, dielectric screening, and intricate fluctuation phenomena distinguish this regime from its Coulomb and log-gas antecedents, and recent advances have provided comprehensive analytic machinery and explicit scaling laws for its equilibrium and dynamical behavior.