Mean-Field Limit and Fluctuations

• Mean-field limit is a framework describing how large interacting systems average out individual behaviors to yield deterministic macroscopic properties.
• Fluctuations around the mean-field limit scale as O(1/√N) and are typically Gaussian, quantifying finite-size corrections and deviations in critical regimes.
• Rigorous methods such as direct N-particle analysis, Fock space techniques, and martingale decompositions underpin the precise estimation of fluctuation effects.

The mean-field limit is a principle underpinning a broad spectrum of statistical and dynamical models, describing the macroscopic behavior of large, interacting systems as the number of constituents tends to infinity. In this regime, each component's interactions are typically rescaled (e.g., by $1/N$ for $N$ particles), so the collective field exerted by all particles becomes deterministic, and individual fluctuations are averaged out. Fluctuations around the mean-field limit quantify the deviations of finite-size systems from this deterministic behavior, often exhibiting universal Gaussian statistics, and are essential for understanding corrections to collective observables, finite-size effects, and critical phenomena.

1. Foundations: Mean-Field Limit and Typical Convergence Theorems

In large interacting systems—ranging from bosonic quantum gases (Lewin et al., 2013), neural networks (Sirignano et al., 2018), spin glasses (Dey et al., 2021), to diffusive particles (Cecchin et al., 1 Sep 2025)—the mean-field limit emerges when the interaction strength per pair is scaled inversely with system size, producing a deterministic macroscopic limit. For bosonic systems, the dynamics of an $N$-boson state under an $O(1/N)$ interaction Hamiltonian converges in norm to a product Hartree state (condensate), governed by a nonlinear Hartree equation. The convergence is typically established via propagation of chaos (asymptotic independence) and law of large numbers for empirical measures: $$\Psi_{N,0} = \sum_{n=0}^N u_0^{\otimes(N-n)} \otimes_s \varphi_{n,0}$$ with the time-evolved state $\Psi_{N}(t)$ decomposing into a leading pure condensate plus fluctuations.

For mean-field spin systems, the collective magnetization undergoes an analogous limiting behavior—deterministically described by self-consistency equations or Fokker–Planck/McKean–Vlasov PDEs (2002.01458, Delgadino et al., 2020), and the empirical magnetization concentrates exponentially fast around the deterministic solution.

For networks, including neural or Hawkes-type processes, mean-field PDEs or functional equations govern the limiting distributions of synaptic activity or membrane potentials (Chevallier et al., 2019, Löcherbach, 2022).

2. Fluctuation Theory: Central Limit Theorems and SPDEs

The leading corrections to the mean-field limit are encoded in fluctuation fields—properly rescaled signed measures or functionals of the system's empirical distributions. Universally, the fluctuations converge (typically at $O(N^{-1/2})$ scale) to solutions of linear, often infinite-dimensional, stochastic partial differential equations (SPDEs):

• Bosonic Fluctuations: In the mean-field regime for interacting bosons, fluctuations orthogonal to the condensate are captured by an effective quadratic Bogoliubov Hamiltonian acting in Fock space (Lewin et al., 2013):

$$\mathbb{H}(t) = \int a^*(x) [h(t) + K_1(t)] a(x)\,dx + \frac12 \iint [K_2(t;x,y) a^*(x)a^*(y) + \overline{K_2(t;x,y)} a(x)a(y)]\,dx\,dy$$

with norm convergence in $\mathcal H^N_s$ and error rates $O(N^{-1/2})$ under higher moment bounds.

• Neural Network Fluctuations: For single-layer networks, the empirical parameter measure's $O(N^{-1/2})$ fluctuations obey a linear SPDE in negative Sobolev space $W^{-J,2}$ (Sirignano et al., 2018):

$$d\eta_t = L_t^*[\rho_t] \eta_t\,dt + \Sigma_t^{1/2} dW_t$$

where $L_t$ is the linearized drift operator, $\Sigma_t$ the noise covariance, and $W_t$ a cylindrical Brownian motion.

• Diffusive Particle Systems: For mean-field interacting SDEs on tori or $\mathbb{R}^d$, the fluctuation process $\rho^N_t = \sqrt{N}(\mu^N_t - \mu_t)$ converges to a solution of a linear SPDE in negative-Sobolev space with functional generator estimates and rates $O(N^{-1/2})$ in test functionals (Cecchin et al., 1 Sep 2025, Chen et al., 2024).
• Spiking Neuron Networks and Hawkes Processes: The measure-valued fluctuations around neural field PDEs (Chevallier et al., 2019) or spike-driven SDEs (Löcherbach, 2022) are also Gaussian and specified by limit SDEs in dual Sobolev spaces, often with explicit covariance operator forms.

The general principle is that, after suitable regularity assumptions and scaling ($N^{-1/2}$), the empirical process' deviations from its deterministic mean-field limit become Gaussian in the infinite-size limit.

3. Critical Phenomena and Non-Gaussian Fluctuations

At critical points, where the mean-field limit undergoes a phase transition (e.g., spontaneous symmetry breaking), the fluctuations can deviate from standard Gaussian behavior:

• Ising and Spin Glass Systems: In the mean-field Ising model at the critical point $(\beta=1,B=0)$, magnetization fluctuations (scaled as $N^{1/4}$) converge to a non-Gaussian law with density $\sim\exp(-x^4/12)$ (Deb et al., 2020).
• Disorder Effects: The presence of quenched disorder (random fields or couplings) can alter universality classes of fluctuations. For example, random field Curie–Weiss models exhibit field-dominated linear critical dynamics, contrasting with non-linear SDEs of homogeneous models (Collet et al., 2011).
• Nonlinear Neural and Diffusive Models: Finite-size corrections near bifurcations, saddle-node points, or in the presence of singular potentials can produce non-Gaussian tails or even type changes in the fluctuation SPDEs (Tsiaze et al., 2013, Delgadino et al., 2020). This reflects subtle interplay between scaling exponents, system specifics, and criticality.

4. Methodologies: Direct N-Particle, Fock Space, Martingale and Functional Methods

The derivation of fluctuation results employs several rigorous methodologies:

• Direct N-Particle Techniques: These analyze the full $N$-body Hilbert space dynamics—employing decomposition into leading and fluctuation sectors via time-dependent unitary maps, norm convergence, and operator inequalities (Lewin et al., 2013).
• Fock Space Methods: Coherent-state expansions, Weyl operator conjugations, and Bogoliubov transformations provide alternative frameworks for quantum fluctuation analysis, though typically require technical localization assumptions and Fock sector projections.
• Ricci–Sobolev and Martingale Decompositions: In neural and diffusive settings, martingale central limit theorems, Skorokhod-space tightness, and analytic PDE techniques (Aldous' criterion, Gronwall bounds) underpin proofs of weak convergence and explicit bounds on the fluctuation fields (Löcherbach, 2022, 2002.01458).
• Replica and Spin Glass Theory: Fluctuations of order parameters (overlaps) in glassy systems are computed via replicated action expansion, RS Hessian diagonalization, and explicit formulas linking thermal and heterogeneous variances to inverse mass parameters (Folena et al., 2022).

5. Model-Specific Fluctuation and Universality Structures

Several canonical models illustrate rich and nuanced fluctuation phenomena:

• Spin Glasses and Mean-Field Ising Models: Multiple fluctuation regimes depend on external field scaling $\alpha$; transitions (at $\alpha=1/4$) divide super-linear, constant, and negligible fluctuations, all displaying Gaussian CLTs with explicit rates and Berry–Esseen bounds (Dey et al., 2021, Deb et al., 2020).
• Spatially Extended Hawkes and Spiking Neural Networks: For spatially extended point processes, fluctuation fields converge to space-time Gaussian processes, governed by linearized neural field evolution and identified noise structure (Chevallier et al., 2019, Chevallier, 2016, Löcherbach, 2022).
• Glass Models with RFOT Phenomenology: The replica method enables explicit computation of overlap fluctuations, delineating intra-state and inter-state variances, applicable to $p$-spin spherical, random orthogonal, and random Lorentz gas models (Folena et al., 2022).
• Mean-Field Matching and Network Fluctuations: Stable matching costs and hub network dynamics in heterogeneous random graphs have mean-field limits and central limit theorems (with critical pseudo-dimension thresholds separating Gaussian and non-Gaussian scaling), offering a robust theory for desynchronization and finite-size corrections (Ahlberg et al., 26 Oct 2025, Bian et al., 2024).

6. Fluctuation Rates, Error Bounds, and Mesoscopic Corrections

Across models, the $O(N^{-1/2})$ scaling of Gaussian fluctuations is generically optimal, with explicit norm convergence rates, Kolmogorov distances, and Berry–Esseen-type effective regularity estimates available (Deb et al., 2020, Cecchin et al., 1 Sep 2025). Mesoscopic expansions (e.g., for neuron potentials) provide second-order corrections, yielding Ornstein–Uhlenbeck laws and accurate approximations even for finite $N$ (Löcherbach, 2022, Klinshov et al., 2015).

Corrections can be more subtle in the presence of singular (e.g., Riesz, Coulomb) kernels, where rates may include logarithmic or algebraic modifications depending on spatial dimension and interaction regime (Chen et al., 2024, Cecchin et al., 1 Sep 2025).

7. Implications: Universality, Critical Behavior, and Future Directions

Fluctuation results clarify finite-size effects, inform uncertainty quantification in computational models, and highlight universality and the breakdown of classical mean-field approximations near critical points or in disordered systems. They underpin the practical accuracy of mean-field approaches for large but finite networks, quantum systems, and statistical ensembles, and provide sharp diagnostics for when corrections or alternative scaling must be invoked.

Continued research explores non-Gaussian corrections in complex environments, non-commuting limits (e.g., mean-field and homogenization in periodic systems (Delgadino et al., 2020)), and finer classification of fluctuation universality classes across equilibrium and non-equilibrium regimes. The general principles surveyed here are essential for the analysis, design, and diagnosis of large interacting systems across statistical physics, probability, and applied mathematics.