Determinantal Coulomb Gas

• Determinantal Coulomb gas is defined by embedding a determinant structure into the Coulomb interaction through Vandermonde factors or fermionic antisymmetry, providing a unified framework across planar, Riemann surface, and one-dimensional settings.
• These models manifest in diverse regimes—from planar log-gases with orthogonal-polynomial kernels to Bergman point processes on compact Riemann surfaces and fermionic one-dimensional quantum gases—each with distinct equilibrium measures and edge behaviors.
• Recent studies leverage precise asymptotic free-energy expansions, local statistical analyses, and exact results under various confinement and fluctuation regimes to deepen our understanding of spectral determinants and conformal functionals.

Determinantal Coulomb gas denotes a class of Coulomb or log-gas models in which the particle law, or the underlying many-body state, is governed by a determinant structure. In the standard planar setting this is the two-dimensional Coulomb gas with joint density proportional to $\prod_{j<k}|z_j-z_k|^{2}\prod_{j=1}^N e^{-NQ(z_j)}$, equivalently the eigenvalue law of random normal matrices. On compact Riemann surfaces, the determinantal case is realized through determinants of holomorphic sections and Bergman-type kernels. In one-dimensional quantum mechanics, the expression is also used for fermionic Coulomb systems whose antisymmetry is determinantal in principle, even when the determinant can be eliminated from the actual analysis by Bose–Fermi mapping (Byun et al., 2022, Bourgoin, 28 Aug 2025, Astrakharchik et al., 2010).

1. Determinantal mechanisms and normalization conventions

The unifying feature is not a single Hamiltonian but a shared structural mechanism: at a special inverse temperature, the Coulomb interaction combines with a Vandermonde factor or a fermionic antisymmetrizer so that correlation functions, partition functions, or wavefunctions can be expressed in determinantal form. In two dimensions this usually means a classical log-gas at the random-matrix temperature; in complex geometry it means a determinantal point process associated with $H^0(M,L)$; in one-dimensional quantum problems it means antisymmetric many-body states whose sign structure is known exactly (Byun et al., 2022, Bourgoin, 28 Aug 2025, Astrakharchik et al., 2010).

The normalization of the inverse temperature is convention-dependent. In several random-normal-matrix papers the determinantal temperature is written as $\beta=2$, while in the hard-wall and Riemann-surface conventions the same determinantal regime is written as $\beta=1$ (Charlier, 31 Jul 2025, Seo, 2020).

Setting	Determinantal object	Representative realization
Planar log-gas	Orthogonal-polynomial kernel	Random normal matrix ensemble
Compact Riemann surface	Determinant of holomorphic sections	Bergman point process
One-dimensional quantum gas	Fermionic antisymmetry / hidden Slater structure	Spinless Coulomb fermions

In the planar case, the determinantal structure is explicit in the joint density. For radially symmetric potentials $Q(z)=q(|z|)$, monomials are orthogonal, the partition function reduces to a product of orthogonal norms, and the kernel is the projection onto the first $N$ orthogonal polynomials (Byun et al., 2022). On compact Riemann surfaces, the partition function at the determinantal temperature is built from the Green function of the Laplacian and an external potential, but bosonization rewrites the Coulomb interaction in terms of $\|\det \omega_i(p_j)\|^2$, analytic torsion, scalar Laplacian determinants, and theta factors (Bourgoin, 28 Aug 2025). In the one-dimensional quantum Coulomb gas, the determinantal aspect is conceptual rather than point-process-theoretic: the fermionic ground state is antisymmetric, but the relevant computations can be performed with a Jastrow–Vandermonde product and an absolute-value mapping to a bosonic problem (Astrakharchik et al., 2010).

2. Equilibrium measures, droplets, and connectivity

For classical two-dimensional Coulomb gases, the macroscopic object is the equilibrium measure minimizing the weighted logarithmic energy. In the radial determinantal ensembles, the minimizer is supported on a droplet that is either a disk or an annulus, and for sufficiently regular $Q$ one has $d\mu_Q(z)=\Delta Q(z)\,\mathbf 1_S(z)\,dA(z)$. The distinction between simply connected droplets and annular droplets is not cosmetic: it governs the subleading free-energy coefficients and the large-$N$ asymptotics of the partition function (Byun et al., 2022).

Hard-wall confinement changes the equilibrium problem qualitatively. For a two-dimensional Coulomb gas localized to a disk $H^0(M,L)$0, if the wall cuts inside the unconstrained droplet, the equilibrium measure acquires a singular component on the wall. In the radially symmetric pushed phase $H^0(M,L)$1, the localized equilibrium measure splits into a bulk density $H^0(M,L)$2 and a boundary mass $H^0(M,L)$3 on $H^0(M,L)$4, with $H^0(M,L)$5 (Seo, 2020). This singular boundary mass is the static origin of several nonstandard edge kernels.

Weak confinement leads to a different macroscopic regime. When the potential grows only like $H^0(M,L)$6 plus a bounded term, the empirical measure still satisfies an LDP with speed $H^0(M,L)$7, but the limiting equilibrium measure need not be compactly supported. The paper on weakly confining 2D Coulomb gases gives explicit determinantal examples: the Cauchy ensemble with equilibrium density $H^0(M,L)$8 on $H^0(M,L)$9, and the spherical ensemble with equilibrium density $\beta=2$0 on $\beta=2$1 (Hardy, 2012). The spherical Coulomb-gas paper reformulates this regime geometrically: Coulomb gases on the Riemann sphere with rotationally invariant potentials and point charges at the poles become planar weakly confining models for which the droplet is the entire complex plane (Byun et al., 13 Jan 2025).

Determinantal Coulomb gases also occur with disconnected droplets. For the potential $\beta=2$2, the droplet is $\beta=2$3. When $\beta=2$4, this set consists of $\beta=2$5 disconnected components, a “lemniscate archipelago,” and insertion of a point charge at the origin deforms kernels without destroying the determinantal structure (Byun et al., 2022).

3. Local statistics and universality classes

The local theory is organized by geometry: bulk, regular boundary, hard edge, pushed hard wall, and singular or disconnected droplets produce different kernels. In a hard-wall disk with radial potential, the pushed phase $\beta=2$6 has a microscopic inward scaling of order $\beta=2$7, rather than the usual $\beta=2$8, and the rescaled determinantal kernel converges to a Laplace-type kernel on the right half-plane,

$\beta=2$9

with $\beta=1$0. In the same regime, the maximal modulus has a Weibull limit law with shape parameter $\beta=1$1 (Seo, 2020). This universality class is distinct from the free-boundary and hard-edge plasma kernels of Ginibre type.

A different interpolation is obtained by boundary-confinement ensembles $\beta=1$2. Here $\beta=1$3 is the free boundary, $\beta=1$4 is the hard edge, and $\beta=1$5 yields ultraweak confinement. In the scaling limit at a regular boundary point, the limiting one-point function is

$\beta=1$6

where $\beta=1$7 is built from a Gaussian convolution involving $\beta=1$8. As $\beta=1$9, the limit field develops a heavy tail and the microscopic distinction between inside and outside becomes ambiguous (Ameur et al., 2019).

Extreme local repulsion is encoded in the smallest-gap process. For the two-dimensional Coulomb gas at determinantal temperature with general potential $Q(z)=q(|z|)$0, the smallest gaps are typically of order $Q(z)=q(|z|)$1, and the point process

$Q(z)=q(|z|)$2

converges to a Poisson point process on $Q(z)=q(|z|)$3 with intensity

$Q(z)=q(|z|)$4

where $Q(z)=q(|z|)$5. Consequently, the $Q(z)=q(|z|)$6-th smallest rescaled gap has density proportional to $Q(z)=q(|z|)$7, with $Q(z)=q(|z|)$8 (Charlier, 31 Jul 2025).

Disconnected droplets do not destroy local edge universality. In the discretely symmetric lemniscate ensembles with a point charge, the macroscopic kernel reflects the skeleton of the droplet and exhibits strong correlations along the boundary, but after $Q(z)=q(|z|)$9 edge scaling each regular boundary component has the same limiting kernel,

$N$0

as the classical Ginibre edge (Byun et al., 2022).

4. Free energy, spectral determinants, and conformal functionals

One of the most developed themes in the subject is the large-$N$1 expansion of the partition function. For radial determinantal Coulomb gases in the plane, the partition functions reduce to products of radial orthogonal norms, and the asymptotic expansions of $N$2 can be computed up to $N$3. In the random-normal-matrix case, the $N$4 and $N$5 terms depend on the connectivity of the droplet: disk and annulus cases have different coefficients, and in the disk case a universal $N$6 term appears. The same paper shows that the planar symplectic, Pfaffian analogue acquires an additional $N$7 contribution involving the logarithmic potential $N$8 (Byun et al., 2022).

On compact Riemann surfaces, the determinantal Coulomb gas admits a bosonization formula that relates the partition function to the magnetic Laplacian, scalar Laplacian, theta functions, and analytic torsion. For the original partition function in the determinantal case, Bourgoin proves an expansion

$N$9

with $\|\det \omega_i(p_j)\|^2$0 and $\|\det \omega_i(p_j)\|^2$1, while $\|\det \omega_i(p_j)\|^2$2 contains $\|\det \omega_i(p_j)\|^2$3. This gives a geometric realization of the Zabrodin–Wiegmann conjecture in the determinantal case and makes the topological dependence explicit through the Euler characteristic (Bourgoin, 28 Aug 2025).

For hard-wall Coulomb gases in a Jordan domain $\|\det \omega_i(p_j)\|^2$4, the partition function at determinantal temperature is

$\|\det \omega_i(p_j)\|^2$5

Its exact expression through the truncated Grunsky operator allows a direct connection with boundary geometry. If $\|\det \omega_i(p_j)\|^2$6 is a Weil–Petersson quasicircle of unit capacity, then

$\|\det \omega_i(p_j)\|^2$7

If $\|\det \omega_i(p_j)\|^2$8 has corners of opening angles $\|\det \omega_i(p_j)\|^2$9, then

$Q$0

The coefficient $Q$1 is therefore the corner anomaly for this determinantal hard-wall gas (Johansson et al., 2023).

Anisotropic quadratic fields with a point charge provide another explicit free-energy laboratory. For

$Q$2

in the regimes where the droplet is simply or doubly connected, the free energy has an expansion through the constant term, with the constant identified with the Liouville action associated with the droplet. This extends the isotropic $Q$3 theory and gives a random-matrix interpretation in terms of moments of characteristic polynomials of the elliptic Ginibre ensemble (Byun et al., 28 May 2026). On the sphere, an analogous program has been carried out for rotationally invariant determinantal and Pfaffian Coulomb gases with point charges at the poles, again yielding precise large-$Q$4 free-energy asymptotics including constant terms (Byun et al., 13 Jan 2025).

5. The one-dimensional quantum Coulomb gas

In one dimension the phrase “determinantal Coulomb gas” has a different status. Astrakharchik and Girardeau study a single-component quantum gas with Hamiltonian

$Q$5

whose fermionic ground state is antisymmetric and therefore determinantal in principle. Their key observation is that the singular $Q$6 repulsion and one-dimensional ordering imply the exact Bose–Fermi mapping

$Q$7

As a consequence, the bosonic and fermionic ground states have identical energies and local observables, and the nodal surface is exactly the set of coincidences $Q$8 (Astrakharchik et al., 2010).

This destroys the usual practical role of the Slater determinant. Instead of evaluating determinants, they use a guiding wavefunction

$Q$9

so that antisymmetry is encoded by a Vandermonde-like factor. Because the nodes are exact, fixed-node diffusion Monte Carlo becomes exact for the fermionic ground-state energy. Because determinants are avoided, the computational complexity is reduced from $d\mu_Q(z)=\Delta Q(z)\,\mathbf 1_S(z)\,dA(z)$0 to $d\mu_Q(z)=\Delta Q(z)\,\mathbf 1_S(z)\,dA(z)$1, which makes $d\mu_Q(z)=\Delta Q(z)\,\mathbf 1_S(z)\,dA(z)$2 feasible in the homogeneous system (Astrakharchik et al., 2010).

The physics crosses between two regimes. At high density the gas approaches the ideal Fermi gas/Tonks–Girardeau limit, with

$d\mu_Q(z)=\Delta Q(z)\,\mathbf 1_S(z)\,dA(z)$3

and phononic low-lying excitations. At low density it approaches a quasi-Wigner crystal with

$d\mu_Q(z)=\Delta Q(z)\,\mathbf 1_S(z)\,dA(z)$4

and the low-lying excitations are plasmons with nonanalytic dispersion

$d\mu_Q(z)=\Delta Q(z)\,\mathbf 1_S(z)\,dA(z)$5

In a harmonic trap, the coarse-grained density changes from a semicircle at weak coupling to an inverted parabola at strong coupling, while the microscopic density develops pronounced shell structure (Astrakharchik et al., 2010).

The one-dimensional model therefore differs sharply from classical planar determinantal point processes. Its determinantal aspect belongs to fermionic antisymmetry, not to a closed kernel calculus for correlation functions, and the main analytical gain comes from removing, rather than exploiting, determinant evaluations.

6. Constraints, fluctuations, and broader extensions

Determinantal Coulomb gases also serve as testbeds for constrained ensembles. For general Coulomb gases under linear or quadratic constraints on the empirical measure, the Gibbs conditioning principle leads to modified variational problems for the equilibrium measure. In the special case of quadratic confinement and a linear constraint, the model is exactly solvable: the conditioning simply shifts the cloud of particles without deformation. More generally, the conditioned equilibrium measure is characterized by an Euler–Lagrange problem with a Lagrange multiplier added to the potential or interaction (Chafaï et al., 2019).

Smooth linear statistics reveal a complementary edge-driven structure. For rotationally invariant Coulomb gases in $d\mu_Q(z)=\Delta Q(z)\,\mathbf 1_S(z)\,dA(z)$6, higher-order cumulants of

$d\mu_Q(z)=\Delta Q(z)\,\mathbf 1_S(z)\,dA(z)$7

can be computed explicitly when $d\mu_Q(z)=\Delta Q(z)\,\mathbf 1_S(z)\,dA(z)$8 is radial. A striking feature is that the higher cumulants depend only on $d\mu_Q(z)=\Delta Q(z)\,\mathbf 1_S(z)\,dA(z)$9 and higher derivatives evaluated at the boundary of the droplet. In the special case $N$0, $N$1, the same formulas can be rederived through the Ginibre determinantal structure (Bruyne et al., 2023).

Disk-counting statistics furnish another exactly determinantal fluctuation problem. For rotation-invariant determinantal processes in the plane, the number of points in $N$2 is representable as a sum of independent Bernoulli variables associated with the eigenvalues of the restricted kernel, and mod-$N$3 convergence yields Berry–Esseen bounds together with precise moderate and large deviation estimates for these counts. The same framework covers the infinite Ginibre process, polyanalytic Ginibre ensembles, Gaussian analytic function zeros, and hyperbolic models (Fenzl et al., 2020).

Taken together, these developments show that “determinantal Coulomb gas” is not a single model but a family of structurally related regimes. In some settings the determinant is the full analytic engine, as in random normal matrices, Bergman point processes, and hard-wall log-gases. In others it survives only as a hidden antisymmetrizer or as a reference point for more general $N$4-ensembles. What persists across these variants is the fusion of Coulomb interaction, conformal or spectral geometry, and exact algebraic structure.