Determinantal Planar Coulomb Gas

• Determinantal Planar Coulomb Gas is a two-dimensional system of repelling charges with joint densities expressed via determinants of a correlation kernel.
• It leverages orthogonal polynomials and equilibrium potential theory to yield explicit results on localization, edge fluctuations (Gumbel statistics), and universal scaling limits.
• The model offers precise asymptotics for the partition function and Gaussian fluctuation properties, linking potential theory with random matrix behavior and geometric analysis.

A determinantal planar Coulomb gas is a two-dimensional system of identical repelling point charges in the complex plane, governed by a joint probability density of determinantal type—characterized by the exact solvability of all $k$-point correlation functions by determinants of a kernel. Such systems are of central interest in random matrix theory and statistical mechanics, serving as universal models for eigenvalues of non-Hermitian random matrices (e.g., the Ginibre ensemble), zeros of certain random analytic functions, and a wide array of two-dimensional particle ensembles with logarithmic interactions. The determinantal structure allows precise, explicit computations of fluctuation, edge, and correlation phenomena. The core mathematical objects are the correlation kernel, orthogonal polynomials, and equilibrium potential theory; the physical phenomena include localization, rigidity, Gumbel statistics at the edge, and universal scaling limits.

1. Definition and Structural Properties

The determinantal planar Coulomb gas is characterized by a joint density for $n$ points $z_1, \ldots, z_n \in \mathbb{C}$: $$P_n(z_1, ..., z_n) = \frac{1}{Z_n} \prod_{j=1}^n e^{-nQ(z_j)} \prod_{1 \leq j < k \leq n} |z_j - z_k|^2,$$ where $Q:\mathbb{C} \to \mathbb{R}$ is an external confining potential, typically rotationally invariant ($Q(z) = V(|z|)$), and $Z_n$ is the partition function normalizing the measure.

Key structural property is the determinantal point process: all $k$-point correlation functions are given by determinants of a kernel $K_n(z,w)$: $$R_n^{(k)}(z_1, ..., z_k) = \det[ K_n(z_i, z_j) ]_{i,j=1}^k,$$ with $K_n(z,w)$ built from a system of planar orthogonal polynomials $\{p_k(z)\}$ with respect to the weight $e^{-nQ(z)}$: $$K_n(z, w) = e^{-n(Q(z)+Q(w))/2} \sum_{k=0}^{n-1} p_k(z) \overline{p_k(w)}.$$ This algebraic structure determines the full (multi-point) distribution of the system.

2. Equilibrium Measure, Droplet, and Localization

The equilibrium measure $\mu_{\mathrm{eq}}$ is the unique minimizer of the weighted logarithmic energy functional

$$I_Q[\mu] = \iint \log \frac{1}{|z-w|} d\mu(z) d\mu(w) + \int Q(z) d\mu(z),$$

over probability measures with compact support. $\mu_{\mathrm{eq}}$ is absolutely continuous with respect to the area measure inside the "droplet" $S = \operatorname{supp} \mu_{\mathrm{eq}}$, typically satisfying $$d\mu_{\mathrm{eq}}(z) = \frac{1}{\pi} \Delta Q(z) \, d^2z \, 1_{S}(z)$$. The scale of the droplet is set by $Q$; for $Q(z) = |z|^2$, $S$ is the unit disk.

Localization results (Ameur, 2019) prove that, with high probability, all particles are within a distance $O(\sqrt{(\log n)/n})$ from $S$, with rapid (Gaussian-type) decay of the one-point function outside, i.e.,

$$R_{n}(z) \leq C n \exp\left( -\beta n \cdot \operatorname{dist}(z, S)^2 \right).$$

This demonstrates strong confinement and absence of particles far from the equilibrium support.

3. Edge Fluctuations and Universality

One of the principal achievements in the theory is the rigorous characterization of second order edge fluctuations ("edge universality"). For radial $Q$, the maximal modulus $|z|_{(1)}$ of the points—after proper centering $b_n$ and scaling $a_n$—has universal Gumbel fluctuations (Chafaï et al., 2013): $P \Big( a_n (|z|_{(1)} - b_n) \leq x \Big) \to \exp(-e^{-x}) \qquad \text{as %%%%24%%%%}.$ For $Q(z) = |z|^\alpha$, explicit formulas are

$$c_n = \log n - 2\log\log n - \log(2\pi), \quad a_n = (\text{const})\sqrt{n c_n}, \quad b_n = (\text{const}) + (\text{const}) \sqrt{c_n/n}.$$

For general smooth, strictly convex $Q$, analogous (explicit) formulas in terms of solutions to $t_0 Q'(t_0)=2$ and $C_0 = [(t_0/2)(2/t_0^2 + Q''(t_0))]^{-1/2}$ exist, confirming universality well beyond the Ginibre ensemble.

The "layered structure" (Chafaï et al., 2013) is a key insight: the radial moduli of the points have the same law as order statistics of independent but non-identically distributed random variables with densities proportional to $t^{2k-1} e^{-nV(t)}$, $k=1,...,n$. This reduction allows Laplace/saddle-point analysis to yield the edge fate.

In settings with weak or inverted confinement, edge particle statistics converge to determinantal point fields defined by the Bergman kernel of the uncharged region (Butez et al., 2018, Ameur et al., 2019), and extremal moduli have non-Gumbel heavy-tailed limits, e.g.,

$$\mathbb{P}\Big( \max_{1\leq i\leq n} |z_i| \leq t \Big) \rightarrow \prod_{k=1}^{\infty} (1-t^{-2k}),$$

for particles escaping into voids due to insufficiently confining $Q$.

4. Fluctuations and Rigidity

At the global scale, linear statistics of the Coulomb gas are rigid and exhibit Gaussian fluctuations with explicit mean and variance (Leblé et al., 2016): $$\operatorname{Fluct}_n(\xi_n) = \int \xi_n\, d\Big( \sum_{i=1}^n \delta_{z_i} - n\mu_{\mathrm{eq}} \Big) \;\xrightarrow{\mathcal{D}}\; N(\mu, \sigma^2)$$

$$\mu = -\frac{1}{8\pi}\int \Delta \xi\, (1_S + (\log \Delta Q)^S), \qquad \sigma^2 = \frac{1}{2\pi \beta} \int |\nabla \xi^S|^2,$$

with $\xi^S$ the harmonic extension outside the droplet. At mesoscopic scales comparable to $n^{-1/2}$, fluctuations remain Gaussian but become mean-zero. Moderate deviation and rigidity estimates show that the variance of linear statistics is $O(1)$ and deviations decay exponentially unless $\tau_n \gg 1$ (i.e., rigidity in the sense of much smaller fluctuations than i.i.d. points).

A further consequence is convergence of the random potential to the Gaussian Free Field (Leblé et al., 2016), up to deterministic shifts: $$\Delta^{-1}\left( \sum_{i=1}^n \delta_{z_i} - n\mu_{\mathrm{eq}} \right) \xrightarrow{\mathcal{D}} \text{GFF}$$ in suitable test-function topology.

5. Partition Function Asymptotics and Topological Terms

Asymptotic analysis of the partition function $\log Z_n$ for radially symmetric potentials, including both simply and doubly-connected ("multi-hole") droplets, recovers the large deviation rate function, explicit subleading terms, and topology-dependent constants (Byun et al., 2022): $$\log Z_n = -N^2 I_Q[\mu_Q] - \frac{1}{2} N \log N + N\left(\frac{1}{2}\log 2\pi - \frac{1}{2} E_Q[\mu_Q]\right) + (\text{topological }\log N+\text{constants}) + \cdots$$ The $O(\log N)$ and $O(1)$ constants depend on whether the droplet is a disk or annulus. For multiply connected droplets (as in the "plasma with holes" (Rougerie, 2 Oct 2025)), there is a universal $(\text{Euler character}) \times \log N$ correction, and free energy variation due to adding or moving holes is independent (to $O(1)$) of location, depending only on topology.

On Jordan domains with corners, the $O(\log N)$ correction acquires a universal "corner anomaly" term (sum over corners of $(\alpha_p + 1/\alpha_p -2)$ where $\alpha_p$ are normalized interior angles) (Johansson et al., 2023). This links the free energy to conformal invariants and the Loewner energy.

On compact Riemann surfaces, the partition function expansion involves the analytic torsion and ultimately verifies the geometric Zabrodin–Wiegmann conjecture in the determinantal case (Bourgoin, 28 Aug 2025).

6. Dynamic and Microscopic Properties

Planar determinantal Coulomb gases can be realized as equilibrium states of interacting Brownian particles with singular (logarithmic) repulsion under a quadratic confining potential (Bolley et al., 2017). In the determinantal regime (inverse temperature $\beta = 2$ or $N^2$-scaling), the joint law matches the Ginibre ensemble. Dynamically, the second moment undergoes Cox–Ingersoll–Ross diffusion, and the system enjoys exponential convergence to equilibrium, Poincaré inequalities, and mean-field propagation.

Microscopically, at low temperatures ($\beta \gg \log n$), particles are uniformly separated and equidistributed (mirroring the zero-temperature Fekete set), with sharp discrepancy estimates (Ameur et al., 2020). The smallest gap scale is $n^{-3/4}$, and the point process of rescaled smallest gaps converges to a Poisson process with intensity controlled by the cubic moment of the equilibrium density (Charlier, 31 Jul 2025).

Edge kernel expansions near the boundary reveal subleading $n^{1/2}$ corrections in the density profile, with coefficients given in potential-theoretic and geometric terms (log-Laplacian derivatives, curvature) (Ameur, 19 Oct 2025). Such refined asymptotics are crucial for understanding extreme statistics, edge rigidity, and universality phenomena.

7. Universality, Scaling Limits, and Outlook

Universality at the edge (Gumbel law for maximum modulus when appropriately confined (Chafaï et al., 2013)), in gap statistics (Charlier, 31 Jul 2025), and for number variances normalized by the local density (Akemann et al., 2022) has been rigorously established in wide classes of determinantal planar Coulomb gases. Scaling limits, such as the appearance of the Bergman or Laplace-type kernel at outliers or hard walls (Butez et al., 2018, Seo, 2020), connote robust, geometry-driven local statistics.

The determinantal structure persists under a wealth of deformations: addition of holes, discrete rotational symmetry, soft or hard boundary conditions, and even on general surfaces (Byun et al., 2022, Byun et al., 2022, Johansson et al., 2023, Byun et al., 13 Jan 2025, Bourgoin, 28 Aug 2025).

This structure underpins a deep interplay between potential theory, random matrix theory, operator determinants, and geometric analysis, and is the controlling framework for universality in complex many-particle systems with two-dimensional logarithmic repulsion.