Linear statistics in the Coulomb gas: cumulants beyond rotational invariance

Technical Summary: Linear Statistics in the Coulomb Gas: Cumulants Beyond Rotational Invariance

Problem Statement
The paper investigates the fluctuations of linear statistics in a classical Coulomb gas of $N$ particles in dimension $d \ge 2$, confined by a general external potential $U$. The particles interact via logarithmic ($d=2$) or Newtonian ($d=3$) repulsion. The primary object of study is the linear statistic $L_N = \sum_{i=1}^N f(x_i)$, where $f$ is a smooth test function.

While the leading-order variance of $L_N$ is well-understood and governed by a central limit theorem with Gaussian fluctuations (scaling as $O(1)$), the behavior of higher-order cumulants for general, non-symmetric potentials and test functions remains largely unexplored. Previous results for higher cumulants (e.g., in [29]) were restricted to rotationally invariant settings where the equilibrium droplet (the support of the particle density) remains a ball under perturbation. This work aims to derive explicit formulas for the leading third and fourth cumulants for general smooth droplets $D_0$ without assuming rotational invariance for either the potential $U$ or the test function $f$.

Methodology
The authors employ a combination of shape calculus, potential theory, and conformal mapping techniques:

1. Tilted Potential and Cumulant Generating Function: The cumulants are derived from the cumulant generating function (CGF) $\chi(s, N) = \log \langle e^{-Ns L_N} \rangle_U$. By introducing a tilted potential $U_s = U + s f / \beta$, the problem is reduced to analyzing the partition function $Z_{N, U_s}$. The $q$-th cumulant is obtained by differentiating $\chi(s, N)$ $q$ times with respect to $s$ at $s=0$.
2. Obstacle Problem and Boundary Velocity: The core of the derivation relies on the connection between the equilibrium measure and the obstacle problem. As the parameter $s$ varies, the support of the equilibrium density (the droplet $D_s$) deforms. The authors utilize the known result that the normal velocity $v_s$ of the droplet boundary $\partial D_s$ is determined by the jump in the normal derivative of the harmonic extension of the test function. Specifically, $v_s = [\vec{\nabla} f_{D_s}] \cdot n_s / (\beta \Delta U_s)$.
3. Shape Derivatives: The authors compute the derivatives of the Dirichlet energy associated with the harmonic extension of $f$ with respect to the domain deformation. This involves calculating the variation of surface integrals over the moving boundary $\partial D_s$, utilizing the Hadamard formula and the properties of the exterior Dirichlet-to-Neumann operator ($\Lambda_+$).
4. Conformal Mapping (d=2): For the two-dimensional case, an independent derivation is provided using complex analysis. This approach utilizes the exterior Riemann map, Julia's variational formula for the evolution of the conformal map under boundary motion, and the Hilbert transform to relate the boundary velocity to the test function.
5. Rigorous Justification (d=2, $\beta=2$): Section 4 provides a rigorous proof of the asymptotic formulas for the third and fourth cumulants in the determinantal case ($d=2, \beta=2$) with real-analytic data. This relies on orthogonal polynomial asymptotics and the bulk Bergman expansion to justify the interchange of limits ($N \to \infty$ and $s \to 0$).

Key Contributions and Results

1. General Third Cumulant Formula:
   The paper derives an explicit formula for the leading third cumulant $\langle L_N^3 \rangle_c$ for any dimension $d \ge 2$ and general smooth droplet $D_0$. The result is expressed as an intrinsic boundary integral over the equilibrium droplet:
   $$\langle L_N^3 \rangle_c \simeq -\frac{1}{N \beta^2 \Omega_d} \int_{\partial D_0} dS \frac{\left( [\vec{\nabla} f_{D_0}] \cdot n \right)^3}{\Delta U}$$
   where $[\vec{\nabla} f_{D_0}] \cdot n$ is the jump of the normal derivative of the harmonic extension of $f$ across the boundary, $n$ is the outward unit normal, and $\Delta U$ is the Laplacian of the external potential (proportional to the background density). The formula explicitly depends on the cube of this jump.

2. General Fourth Cumulant Formula:
   The fourth cumulant is derived as a boundary integral involving additional geometric and spectral operators:
   $$\langle L_N^4 \rangle_c \simeq -\frac{1}{\beta^3 \Omega_d N^2} \int_{\partial D_0} dS \left[ 3 \frac{K^2}{\Delta U} \Lambda_+ \left( \frac{K^2}{\Delta U} \right) + 4 \frac{K^3 \Delta f}{(\Delta U)^2} + 2M \frac{K^4}{(\Delta U)^2} + \frac{K^4 \partial_n \Delta U}{(\Delta U)^3} \right]$$
   where $K = [\vec{\nabla} f_{D_0}] \cdot n$, $\Lambda_+$ is the exterior Dirichlet-to-Neumann operator, $M$ is the mean curvature of the boundary, and $\partial_n \Delta U$ is the normal derivative of the background density. Unlike the third cumulant, the fourth involves the action of the Dirichlet-to-Neumann operator on the square of the jump, as well as curvature and density gradient terms.

3. Special Geometries:
   The general formulas are specialized to specific geometries, including:

   • Disk ($d=2$): Expressed in terms of the Hilbert transform and Fourier modes.
   • Half-plane ($d=2$): Simplified to involve the Hilbert transform of the derivative of $f$.
   • Sphere ($d=3$): Expressed using spherical harmonics and the exterior Dirichlet-to-Neumann operator eigenvalues.
     These specialized results recover known radial formulas when $f$ is rotationally invariant.

4. Validation:
   The authors verify their general formulas against exactly solvable non-radial deformations:

   • Translated elliptic droplets in $d=2$ (Appendix A).
   • Spheroidal deformations of a ball in $d=3$ (Appendix B).
     In the case of $d=2, \beta=2$, the results are shown to match the formulas proposed by Wiegmann and Zabrodin [51], providing a rigorous proof for the third and fourth cumulants in that specific setting.

Significance and Claims
The paper claims to provide the first explicit formulas for the leading third and fourth cumulants of smooth linear statistics in Coulomb gases for general, non-rotationally invariant potentials and test functions in arbitrary dimensions $d \ge 2$.

• Beyond Gaussian Fluctuations: While the variance is well-known to be $O(1)$ and Gaussian, these higher cumulants represent the first non-Gaussian corrections. The derived formulas explicitly show the scaling factors $N^{-1}$ and $N^{-2}$ in the integrands, consistent with the behavior observed in rotationally invariant cases, but the primary novelty lies in the complex integral structure involving boundary jumps, Dirichlet-to-Neumann operators, and curvature.
• Intrinsic Boundary Dependence: A key finding is that these higher cumulants depend only on the boundary values of the test function and its derivatives, as well as the geometry of the droplet boundary (curvature, normal derivatives of the density). This extends the "edge universality" observed in the variance to higher orders.
• Rigorous Foundation: For the case $d=2, \beta=2$, the paper offers a rigorous derivation that validates previous heuristic results from the physics literature (Wiegmann and Zabrodin).
• Limitations: The authors note that the general derivation assumes the perturbed droplets remain smooth and do not undergo topological transitions (e.g., splitting or merging). The rigorous proof for general $d$ and $\beta$ relies on a sufficient condition regarding the derivatives of the partition function (Remark 4.6), which is not fully proven for all cases but is established for the determinantal case.

The work bridges statistical mechanics, random matrix theory, and free boundary problems (obstacle problems), offering a unified framework for understanding non-Gaussian fluctuations in interacting particle systems beyond symmetric configurations.