Spectral gaps for mean-field Coulomb gases

Technical Summary: Spectral Gaps for Mean-Field Coulomb Gases

Problem Statement
The paper addresses the quantitative understanding of the relaxation to equilibrium for classical $N$-particle overdamped Langevin dynamics featuring quadratic confinement and repulsive Coulomb interactions in dimensions $d \ge 3$. The central question is whether the spectral gap of the associated generator (which governs the exponential rate of convergence to the invariant measure) can be controlled uniformly as the number of particles $N$ tends to infinity.

While significant progress has been made for regular mean-field systems and specific singular models (e.g., in $d=2$ or for log-potentials), a general quantitative result for Coulomb gases in $d \ge 3$ with weak coupling has remained incomplete. The authors specifically investigate the regime where the inverse temperature $\beta_N$ may depend on $N$, focusing on the "weak-coupling" condition where the effective interaction strength seen by a single particle remains small.

Methodology
The authors employ a multi-step strategy combining functional analysis, probabilistic coupling, and spectral theory:

1. Scaling and Reduction to One-Site Measures:
   The system is rescaled to transform the equilibrium measure into a form where the quadratic confinement is fixed (Gaussian) and the interaction strength is captured by a parameter $b_N$. The conditional law of a single particle, given the positions of all others, is identified as a Gaussian measure tilted by $N-1$ Coulomb poles. The total charge of these poles is denoted by $\Theta_N$.

2. Uniform One-Site Poincaré Inequality:
   A core technical contribution is the proof of a Poincaré inequality for the "one-site" measure (a Gaussian perturbed by an arbitrary finite number of Coulomb poles).

   • Ground-State Transform: The authors utilize a singular ground-state transform to map the Dirichlet form of the Langevin generator to the quadratic form of an associated Witten Laplacian.
   • Handling Singularities: Crucially, they demonstrate that the singularities at pole locations (collisions) do not contribute additional terms to the closed form. This is achieved by showing that removing poles via cutoff functions incurs zero weighted form cost.
   • Compactness Argument: Using the harmonicity of the Coulomb potential ($\Delta |x|^{2-d} = 0$ away from the origin) and weak-type estimates on the Coulomb field, they establish a compactness argument. This shows that as the total charge of the poles tends to zero, the spectral gap of the perturbed measure converges to that of the Gaussian oscillator, which is strictly positive. This yields a Poincaré constant uniform in the number and positions of the poles, provided the total charge is sufficiently small.

3. Dobrushin-Wu Tensorization:
   To extend the one-site inequality to the full $N$-particle system, the authors apply the Dobrushin-Wu tensorization theorem.

   • They define a Dobrushin interdependence matrix $C_N$ based on the total variation distance between conditional laws when a single boundary coordinate is changed.
   • Using the moving-pole estimates derived from the one-site analysis, they bound the entries of $C_N$.
   • Under the assumption that the effective interaction strength $\Theta_N$ is sufficiently small, the spectral radius $\rho(C_N)$ is strictly less than 1. This condition allows the tensorization of the uniform one-site Poincaré inequalities to yield a uniform lower bound for the many-particle system.

4. Upper Bound:
   The upper bound is derived using the center-of-mass coordinate. Since the Coulomb interaction depends only on particle differences, it is invariant under translation of the center of mass. The Gibbs measure factorizes exactly in this coordinate, allowing the construction of a symmetric test function (a component of the center of mass) that achieves the upper bound.

Key Results
The main result (Theorem 1.1) establishes that for dimensions $d \ge 3$, there exists a threshold $\varepsilon_d > 0$ such that if the effective interaction strength $\Theta_N$ satisfies $\limsup_{N \to \infty} \Theta_N < \varepsilon_d$, the spectral gap $\lambda_N^d(\beta_N)$ is uniformly bounded from below and above:
$$c_{d, \chi, (\beta_N)} \frac{\alpha_N}{N} \le \lambda_N^d(\beta_N) \le 2 \frac{\alpha_N}{N}$$
where $\alpha_N$ is the reversible speed (typically $\alpha_N = N$).

Specifically:

• Uniformity: The lower bound constant is independent of $N$, provided the weak-coupling condition holds.
• High-Temperature Regime: The condition $\Theta_N < \varepsilon_d$ corresponds to a high-temperature or weak-coupling regime. If the physical inverse temperature $\beta_N/N$ remains bounded, the condition simplifies to $\chi (\beta_N/N)^{d/2} < \varepsilon_d$.
• Labelled vs. Unlabelled: The results hold for both labelled (non-symmetric) and unlabelled (symmetric) spectral gaps, as the upper bound test function is symmetric and the lower bound holds for the larger class of all functions.

Significance and Claims
The paper claims to provide a rigorous, quantitative proof of uniform exponential relaxation for mean-field Coulomb gases in dimensions $d \ge 3$ under weak coupling.

• Overcoming Non-Convexity: The work bypasses the failure of standard Bakry-Émery arguments (which require global convexity) by utilizing the specific homogeneity and harmonicity of the Coulomb potential combined with a compactness argument for the ground-state transform.
• Handling Collisions: A significant technical achievement is the rigorous treatment of particle collisions (coincident poles), proving that the singular nature of the interaction does not destroy the spectral gap or the validity of the Poincaré inequality.
• General Framework: The authors note that while the specific treatment of $d=2$ (log-potential) was omitted due to algebraic differences, the general strategy suggests similar results could be obtained for $d \ge 3$ log-potentials by replacing compactness arguments with Hardy-Muckenhoupt estimates, though this generalization is not pursued in the text.

The paper does not propose new experimental applications or future directions beyond the mathematical framework established, focusing strictly on the derivation of the spectral gap bounds for the specified stochastic dynamics.