Mean-Field Log-Gases: Dynamics & Universality

• Mean-field log-gases are interacting particle systems characterized by long-range logarithmic repulsion and external potentials, central to random matrix theory.
• They employ advanced techniques like modulated free energy methods and functional inequalities to precisely describe equilibrium, fluctuations, and propagation of chaos.
• Their study uncovers universal features such as determinantal structures and phase transitions, with implications for statistical physics and large deviation analysis.

A mean-field log-gas is an interacting particle system in which each particle experiences long-range logarithmic repulsion from all other particles, possibly subjected to additional confining or mean-field interactions. The fundamental structure arises in random matrix theory, statistical physics, and probability, particularly as the β-ensemble and its dynamical and thermodynamic generalizations. Mean-field log-gases provide a universal paradigm for strongly correlated systems, encode connections to determinantal point processes, concentration inequalities, universality in random matrix statistics, propagation of chaos, functional inequalities, and large deviation principles. Advances in the analysis of these systems focus on establishing quantitative behavior in the large particle number limit, deriving precise equilibrium and fluctuation results, and clarifying the role of convexity, semi-convexity, dimension, and temperature.

1. Mathematical Formulation and Structure

The canonical mean-field log-gas on a domain $A \subseteq \mathbb{R}^d$ for $N$ particles has the joint law (high-dimensional Gibbs measure)

$$m_N(x^1,\ldots,x^N) = \frac{1}{Z_N} \exp\left(-\beta \left[ \sum_{i \neq j} g(x^i-x^j) + N\sum_{i=1}^N V(x^i) \right]\right),$$

where $g(x) = -\log|x|$ in $d=1,2$ (logarithmic repulsion) and $g(x)$ can be the Coulomb kernel $|x|^{2-d}$ for $d \geq 3$. Here $V$ is a confining external potential, and $\beta$ is the inverse temperature.

• The empirical measure $\mu_N = \frac{1}{N} \sum_{i=1}^N \delta_{x^i}$ converges (as $N \to \infty$) to the equilibrium measure $\mu_V$ minimizing the mean-field energy:

$$\mathcal{I}_V(\mu) = \iint g(x-y)\, d\mu(x) d\mu(y) + \int V(x) d\mu(x)$$

subject to probability constraints.

• Fluctuations and beyond-mean-field corrections are encoded through the decomposition of the energy into macroscopic, confinement, and fluctuation (microscopic) terms:

$$H_N = N^2\, \mathcal{I}_V(\mu_V) + 2N \sum_{i=1}^N \zeta(x_i) + F_N^V$$

with $F_N^V$ involving the renormalized energy of fluctuations away from equilibrium (Serfaty, 2017).

Renormalized energies at the microscopic scale, and their connections to "electric potentials," provide tools for describing crystallization and phase transitions.

2. Mean-Field and Hydrodynamic Limits

In the dynamic regime, each particle's evolution is governed by an SDE

$$dX_t^i = -\nabla V(X_t^i)\, dt - \frac{1}{N}\sum_{j \neq i} \nabla_1 W(X_t^i, X_t^j)\, dt + \sqrt{2}\,dW_t^i,$$

with $W(x,y) = -\log|x-y|$ for the log-gas (Delgadino et al., 27 Jun 2025). For large $N$ and suitable initial data, the empirical process converges to the McKean–Vlasov equation, e.g.

$$\partial_t \bar\rho_t = \Delta \bar\rho_t + \nabla \cdot [ \bar\rho_t ( \nabla(W*\bar\rho_t) + \nabla V ) ]$$

(Unterberger, 2016, Cai et al., 22 Nov 2024).

Quantitative propagation of chaos has been established in relative entropy (Cai et al., 22 Nov 2024), i.e. for $N$-particle law $\rho^N_t$ and product measure $\bar \rho_t^{\otimes N}$,

$$\overline H(\rho^N_t \mid \bar \rho_t^{\otimes N}) \leq C e^{Ct} \left( \overline H(\rho^N_0 \mid \bar \rho_0^{\otimes N}) + \frac{1}{N} \right)$$

with further refinement showing the optimal $O(1/N)$ scaling for the logarithmic interaction (removing previous $\log N / N$ losses) (Delgadino et al., 27 Jun 2025).

These analyses employ modulated free energy methods, exploiting the partition function's uniform bound with respect to $N$ by adapting techniques from Euclidean QFT (Nelson's construction) to control singularity (Delgadino et al., 27 Jun 2025).

3. Universality, Determinantal Structure, and Random Matrices

Mean-field log-gas ensembles, such as

$$\mu_{\beta,V}^{(N)}(\lambda) \sim \exp\left[-\beta N\left( \sum_{j=1}^N \frac{1}{2}V(\lambda_j) - \frac{1}{N} \sum_{i<j} \log|\lambda_i-\lambda_j| \right)\right],$$

naturally arise as eigenvalue laws of invariant random matrices (Wigner–Dyson–Mehta ensembles) (Erdos, 2012, Erdos, 2014).

Key universality phenomena:

• Local eigenvalue statistics, after microscopic rescaling, depend only on the symmetry parameter $\beta$ and not on $V$ or the entry distribution ("Wigner–Dyson–Mehta universality").
• Gap distributions in the bulk converge to those of the sine kernel point process (for $\beta=2$) and more generally $\beta$-ensembles exhibit determinantal or Pfaffian correlation structure (Katori, 2020, Wolff et al., 2021).
• Rigorous proofs leverage the Dyson Brownian motion as a local ergodicity mechanism, the local semicircle law, and De Giorgi–Nash–Moser regularity theory to show Hölder continuity for gap observables (Erdos, 2012, Erdos, 2014).
• Scaling limits at the edge (e.g., the largest eigenvalue) yield Tracy–Widom statistics (soft-edge universality) via asymptotics of Fredholm determinants governed by correlation kernels (Wolff et al., 2021).

These results extend to more realistic models, including random band matrices and dilute mean-field graphs, provided suitable connectivity conditions hold (Erdos, 2012, Bauerschmidt et al., 31 Mar 2025).

4. Functional Inequalities, Log-Sobolev Bounds, and Concentration

Functional inequalities are central to controlling long-time dynamics and concentration of measure in mean-field log-gases:

• Under strong log-concavity (Bakry–Émery criterion), the Gibbs law satisfies a uniform log-Sobolev inequality (LSI), implying rapid mixing and sub-Gaussian concentration (Lacker et al., 2022, Chewi et al., 16 Sep 2024).
• For flat-convex or semi-convex energy functionals $F$ (convex along affine interpolations in Wasserstein space), uniform-in-$N$ LSIs and Poincaré inequalities are established, using conditional functional inequalities and $\Gamma_2$ calculus (Wang, 6 Aug 2024, Monmarché, 26 Sep 2024). These results extend to small negative curvature provided uniform constants dominate the negative part.
• In the weak/soft convexity regime, generalized HWI inequalities control convergence in Wasserstein distance, permitting explicit exponential decay rates even with double-well or nonconfining potentials, provided tails are controlled (Mustapha, 2020).
• For high temperatures (small $\beta$), the log-Sobolev constant can remain positive up to the critical point (phase transition), while at criticality it decays to zero, with explicit scaling in $(T-T_c)$ for double-well cases (Bauerschmidt et al., 31 Mar 2025).
• Uniformity in $N$ of these inequalities ensures that rates do not deteriorate with increasing system size, underpinning propagation of chaos and mean-field approximations (Chewi et al., 16 Sep 2024, Wang, 6 Aug 2024).

5. Large Deviations, Fluctuations, and Renormalization

Mean-field log-gases exhibit precise Gaussian-type bulk fluctuations and large (non-Gaussian) deviations at the edge:

• Central limit theorems for linear statistics apply at leading (macroscopic) scale, with variance and mean given explicitly by the solution to loop equations (Schwinger–Dyson equations) and their $1/N$ expansion (Borot et al., 2013, Serfaty, 2017).
• At finer scales, the system satisfies a large deviations principle (LDP) at speed $N$ for empirical fields, with a rate function given by the sum of renormalized microscopic energy and entropy terms:

$$\bar{\mathcal{F}_\beta}(\bar{P}) = \frac{\beta}{2}\, \bar{\mathbb{W}}(\bar{P}, \mu_V) + \bar{\operatorname{Ent}}(\bar{P} |\Pi^1)$$

(Serfaty, 2017).

• Edge LDPs: At high temperature (where $\beta_N \to 0$ as $N \to \infty$), the maximum particle or leading eigenvalue exhibits deviations on the scale $(\log N)^{1/\alpha}$, with a rate function matching that of i.i.d. particles:

$$I_\alpha(x) = \begin{cases} x^\alpha - 1 &\text{if } x \geq 1,\ +\infty &\text{otherwise} \end{cases}$$

for confining potentials $V(x) \sim |x|^\alpha$ (Guera et al., 18 Jul 2025). For tridiagonal random matrices with Gaussian tails, the rate function is $I_2(x) = x^2 - 1$ (Guera et al., 18 Jul 2025). Deviations are typically governed by atypical events involving only a few particles or entries.

• Beyond mean-field: Renormalized energies at the microscopic scale rigorously connect mean behavior with crystallization and statistical fine structure (Serfaty, 2017).

6. Partition Function Asymptotics and Topological Expansions

The partition function $Z_N$ of the mean-field log-gas admits an all-order asymptotic expansion: $$Z_N = N^{\alpha N} \exp\left\{ \sum_{k=-2}^{k_0} N^{-k} F^{[k]} + o(N^{-k_0}) \right\} \Theta(v; T)$$ with $F^{[k]}$ smooth functions of the filling fractions and potential, and a pseudo-periodic $\Theta$-function encoding multi-cut or moduli dependence (Borot et al., 2013). The expansion schemes rely on analyticity, off-criticality, and strict convexity (to control fluctuations and ensure concentration). The emergence of topological recursion structures connects to matrix models and enumerative geometry.

7. Extensions, Generalizations, and Open Problems

• Generalizations include systems with multicomponent charges, non-quadratic or weakly coupled interactions, and dilute or graph-based mean-field structure (Wolff et al., 2021, Bauerschmidt et al., 31 Mar 2025).
• Determinantal martingale frameworks capture both spatial and temporal correlations and allow for systematic calculation of dynamical (non-equilibrium) fluctuation properties (Katori, 2020).
• Integration with conformal field theory arises for processes on planar domains through coupling with Gaussian free fields and SLE (Katori, 2020).
• Sharp $1/N$ convergence rates in the modulated free energy approach can be achieved even for singular (logarithmic) repulsion, a result that uses analogies with exponential integrability and renormalization from Euclidean QFT (Delgadino et al., 27 Jun 2025).
• Remaining open directions include extensions to higher dimensions, time-uniform propagation of chaos, precise control at critical points for phase transitions, and the connection to universality in disordered and non-mean-field systems.

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The paper of mean-field log-gases thus unifies deep tools from probability, analysis, and mathematical physics—delivering rigorous insight into the statistical mechanics of long-range interacting systems, ground state organization, universality classes, and non-equilibrium evolution at both macroscopic and microscopic scales.