Local Central Limit Theorem (LCLT)

• LCLT is a refinement of the classical CLT that delivers pointwise approximations of probability masses and densities using Gaussian profiles.
• It is applied in lattice systems, interacting spin models, stochastic processes, and random matrix theory using techniques like Fourier analysis and cluster expansions.
• Recent advances extend the LCLT to non-Markovian and functional settings with explicit error bounds and quantitative rate estimates.

The local central limit theorem (LCLT) is a fundamental result in probability theory and statistical mechanics, refining the classical central limit theorem (CLT) by providing precise asymptotic approximations for the probability mass (or density) at specific points or in small intervals. Unlike the CLT, which describes convergence in distribution to the normal law at macroscopic (integral or cumulative) scales, the LCLT delivers pointwise or fine-scale approximations, typically by matching normalized probabilities to the Gaussian density up to vanishing errors. The LCLT plays a central role across lattice systems, stochastic processes, random matrix theory, combinatorics, geometry, and statistical physics, and has seen recent advances in interacting systems with long-range, unbounded, or non-Markovian dependencies.

1. Abstract Formulations and Fundamental Principles

The local central limit theorem asserts, under appropriate normalization and regularity assumptions, that for a sequence of random variables (often sums or functionals of many weakly dependent pieces), the (properly rescaled) probability mass at or near a fixed value converges to the standard normal density. In a standard lattice case, for variables $S_n$ with mean $m_n$ and variance $D_n\to\infty$,

$$\sup_{k\in\mathbb{Z}} \left| \sqrt{D_n}\, P(S_n = k) - \frac{1}{\sqrt{2\pi}}e^{-\frac{(k-m_n)^2}{2D_n}} \right| \xrightarrow{n\to\infty} 0.$$

In the continuous setting, for densities $p_n(u)$,

$$\sup_{u\in\mathbb{R}} \left| \sqrt{D_n}\,p_n(u) - \frac{1}{\sqrt{2\pi}} e^{-\frac{(u-m_n)^2}{2D_n}} \right| \xrightarrow{n\to\infty} 0.$$

Extensions to vector-valued and functional settings involve convergence in appropriate metrics and normalization of multivariate densities.

The LCLT requires more stringent conditions than the CLT: typically aperiodicity (lattice), nonlattice structure (continuous), regularity/decay of higher moments, and sometimes mixing or independence conditions.

2. LCLTs in Interacting Spin Systems and Statistical Mechanics

Recent breakthroughs have established equivalence between the integral CLT and the LCLT for a broad range of lattice spin systems with long-range interactions, both for bounded and certain classes of unbounded spins.

For systems with spins on $\mathbb{Z}^d$, pairwise interactions $J_{xy}$ satisfying absolute summability $\sum_{y} |J_{xy}|<\infty$, and generic a-priori single-site measures, the following holds:

• Bounded spins: For any temperature and arbitrary boundary conditions, the integral CLT for the total spin (normalized sum) is equivalent to the LCLT. The proof leverages Fourier-analytic inversion and decimation (dilution) techniques combined with cluster expansions, allowing uniform control of the characteristic function across relevant frequency bands by partitioning integration into "small," "medium," and "large" frequency regimes (Procacci et al., 2023, Endo et al., 8 Aug 2024).
• Unbounded spins: Under strong boundary conditions ("tempered") and sufficiently high temperature, cluster expansion techniques establish uniform bounds required for the LCLT. The main difference with the bounded case lies in the control of the large-$|\sigma_x|$ tail, necessitating high temperature for convergence of the expansion (Endo et al., 2021, Endo et al., 8 Aug 2024).
• Extension to finite and infinite volumes: All results are uniform in the boundary condition, apply to non-translation-invariant, infinite-range interactions, and, for bounded spins, require no temperature restrictions.
• Methodology: The Fourier-analytic approach proceeds by partitioning the characteristic function's integration domain and employing cluster expansions (polymer methods), decimation arguments to induce effective high-temperature conditions, and uniform cumulant and exponential decay estimates (see §3–7 of (Endo et al., 8 Aug 2024, Procacci et al., 2023)).

3. LCLTs in Dynamical Systems and Flows

Suspension (semi-)flows constructed over probability-preserving maps with "roof" functions admit an LCLT for ergodic sums under a small set of abstract, verifiable hypotheses:

• Key Hypotheses: (i) Base-map LCLT (joint distribution of observable and roof), (ii) moderate deviations, (iii) sufficient mixing of the flow (exponential or stretched-exponential), (iv) nonlattice (minimal closed group generated by observable/roof is $\mathbb{R}^2$), (v) regularity (bounded, Hölder, spectral-gap for transfer operator).
• Examples: Axiom A flows, Sinai billiard with finite horizon, suspensions over Pomeau-Manneville maps with tail exponent $>2$, geometric Lorenz attractors. In each, variance is explicitly given (Green-Kubo), and lattice/nonlattice dichotomy is treated via a Haar measure factorization (Dolgopyat et al., 2017).
• Proof Strategy: Decompose ergodic sum into base sums and remainders; use base LCLT and mixing/spectral-gap for transfer operators; sum over possible discrete time levels to recover the continuous limit.

4. Quantitative and Combinatorial LCLTs

Sharp (explicit-rate) local limit theorems have been developed for sums of independent, perturbed log-concave integer-valued random variables—covering rich combinatorial objects:

• Quantitative LCLT: For weighted sums $W_n = \sum_{k=1}^n kX_k$ with independent $C$-perturbed-log-concave $X_k$, the LCLT holds with an explicit bound: $$\sup_m \left| \frac{\sigma}{h}P(W_n=m) - \phi\left(\frac{m-\mu}{\sigma}\right) \right| \leq C_6 \left( \frac{\sigma_{\max}}{\sigma} + C_5 \sigma e^{-C_4|M|} \right),$$ where $\sigma$, $\sigma_{\max}$, and $M$ capture scaling and variance structure (DeSalvo et al., 2016).
• Combinatorial applications: LCLTs enable precise asymptotic enumeration for integer partitions, set partitions (Bell numbers), assemblies, multisets, and selections, with explicit error rates, and under mild regularity conditions (perturbed log-concavity).
• Arratia–Tavaré Principle: This LCLT framework provides detailed total variation error estimates for component counts in random combinatorial structures via conditioning arguments.

5. LCLTs in Stochastic Processes and Geometry

LCLT methodology extends to Markov chains with spectral gap, random walks on expander graphs, and various models in stochastic geometry:

• Markov Chains and Expanders: For functionals of random walks on expander graphs, the LCLT holds with uniform $O(1/t)$ errors, attributable to spectral gap and decomposition into nearly independent increments (Chiclana et al., 2022).
• Stochastic Geometry: The general two-component LCLT framework allows derivation of local limits for percolation cluster sizes, number of components in random graphs, accepted points in random sequential adsorption, and nearest-neighbour functionals. The method isolates a large "LCLT-part" and treats the remainder via a classical CLT, propagating local limits to the global quantity (Penrose et al., 2010).
• Refined Corrections: For random walks with local defects or perturbations (e.g., perturbation at the origin), explicit short- and long-range corrections to Gaussian leading behavior are calculable via Fourier inversion, with dimension-dependent structure (Genovese et al., 2016).

6. LCLTs in Random Matrix and Point Processes

The LCLT has been rigorously established for counts in determinantal point processes and for spectral statistics of classical and non-Hermitian random matrix ensembles:

• Determinantal processes: For the number of points in growing domains for determinantal processes with Hermitian kernels (including bulk and edge scaling for GUE, Ginibre, and GSE), the LCLT holds whenever the variance diverges (Forrester et al., 2013). This follows from the Fredholm determinant expansion, log-concavity stemming from Lee–Yang theory, and Bender’s theorem.
• Elliptic GinOE: For the number of real eigenvalues in the elliptic Ginibre orthogonal ensemble, the LCLT is established by showing the real-zero property for the generating function, variance divergence, and application of classical log-concavity results (Forrester, 2023).
• Quantitative bounds: Uniform quantitative rates ($O(1/\sigma_N^2)$) in $N$ have been derived for the real eigenvalue count in these ensembles.
• Universality: The LCLT provides strong evidence for universality in local fluctuations (e.g., spacings, extremal eigenvalues), with applications in extreme value theory and level statistics.

7. Variations, Extensions, and Connections

• Mixing LCLTs: In dynamical systems, the mixing local CLT (MLCLT) allows insertion of arbitrary "past" and "future" observables, providing a bridge between local pointwise statistics and global ergodic properties. This is established for full-branch expanding interval maps with unbounded, oscillatory observables, including applications to the value distribution of the Riemann zeta function (Fernando et al., 2023).
• Reflected diffusions and random environments: For symmetric diffusions in stationary, possibly degenerate random environments (including diffusions on continuum percolation clusters), quenched LCLTs are deduced from parabolic Harnack inequalities, ergodicity, and invariance principles, with explicit control of transition densities (Chiarini et al., 2015, Takeuchi, 2023).
• Equivalence with integral CLT: Under absolute summability and bounded spin, the equivalence of LCLT and CLT is robust (see (Endo et al., 8 Aug 2024, Procacci et al., 2023, Endo et al., 2021)), extending to general pair potentials and a wide array of spatially extended models.
• Multivariate and functional extensions: LCLT formulations extend for joint distributions of vector-valued observables (such as multi-group Curie-Weiss models), yielding multivariate Gaussian approximations at the local scale (Fleermann et al., 2020).

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References:

• (Endo et al., 8 Aug 2024): Local Central Limit Theorem for unbounded long-range potentials
• (Procacci et al., 2023): On the local central limit theorem for interacting spin systems
• (Dolgopyat et al., 2017): On mixing and the local central limit theorem for hyperbolic flows
• (Chiclana et al., 2022): A local central limit theorem for random walks on expander graphs
• (Genovese et al., 2016): Local Central Limit Theorem for a Random Walk Perturbed in One Point
• (DeSalvo et al., 2016): A robust quantitative local central limit theorem with applications to enumerative combinatorics and random combinatorial structures
• (Forrester et al., 2013): Local Central Limit Theorem for Determinantal Point Processes
• (Forrester, 2023): Local central limit theorem for real eigenvalue fluctuations of elliptic GinOE matrices
• (Penrose et al., 2010): Local central limit theorems in stochastic geometry
• (Endo et al., 2021): Local Central Limit Theorem for Long-Range Two-Body Potentials at Sufficiently High Temperatures
• (Chiarini et al., 2015): Local Central Limit Theorem for diffusions in a degenerate and unbounded Random Medium
• (Fleermann et al., 2020): Local Central Limit Theorem for Multi-Group Curie-Weiss Models
• (Fernando et al., 2023): Limit Theorems for a class of unbounded observables with an application to "Sampling the Lindelöf hypothesis"
• (Takeuchi, 2023): Local Central Limit Theorem for Reflecting Diffusions in a Continuum Percolation Cluster