Central Limit Theorem for continuous $N$-particle systems in fixed- and high-temperature: single level

Technical Summary: Central Limit Theorem for Continuous N-Particle Systems in Fixed- and High-Temperature

Problem Statement
This paper addresses the characterization of Central Limit Theorems (CLTs) for polynomial linear statistics of continuous $N$-particle systems on $\mathbb{R}$. The study covers two distinct asymptotic regimes:

1. Fixed-temperature: The inverse temperature parameter $\theta = \beta/2 > 0$ is fixed while $N \to \infty$.
2. High-temperature: The interaction weakens as $N$ grows, such that $\theta_N \to 0$ and $N\theta_N \to \gamma > 0$.

The systems are modeled via a Bessel generating function (BGF), $G_{N,\theta}(\vec{x}) = \mathbb{E}_{\mu_N}[B_{\vec{a}}(\vec{x}; \theta)]$, where $B_{\vec{a}}$ is the multivariate Bessel function associated with the Jack parameter $\theta$. The goal is to determine necessary and sufficient conditions for the CLT to hold in terms of the coefficients of the logarithmic BGF and to derive explicit combinatorial formulas for the limiting covariance.

Methodology
The proof strategy combines operator-theoretic moment extraction with a topological expansion of the multivariate Bessel function.

1. Dunkl Operators and Moment Extraction:
   The author utilizes Dunkl operators $D_i$ to extract moments from the BGF. By applying these operators to the logarithm of the BGF, $\log G_{N,\theta}$, the covariance of polynomial observables is expressed as a sum over weighted configurations. These configurations are modeled as Lukasiewicz paths (random walks with specific step constraints) equipped with block functions that track the distribution of degrees among variables.

   • In the fixed-temperature regime, the leading order terms cancel, and the covariance is determined by contributions invisible to the Law of Large Numbers (LLN).
   • The analysis involves classifying these configurations into "first-order" (involving only first-order free cumulants) and "second-order" (involving second-order free cumulants) based on the number of specific operator types (Type II and Type V) acting on the path.

2. Topological Expansion:
   To establish the "only if" direction (necessity) and the vanishing of higher-order cumulants, the paper employs a topological expansion of $\log B_{\vec{a}}(\vec{x}; \theta)$. This expansion, derived from Jack series identities (Chapuy–Dołęga), expresses the logarithm as a sum over constellations (ribbon graphs embedded on surfaces).

   • Each term is weighted by a factor involving the genus $g$ of the underlying surface and a non-orientability statistic $\nu_\rho$.
   • Fixed-temperature: The scaling implies that only genus-zero (planar) constellations contribute to the leading order.
   • High-temperature: The scaling $N\theta \to \gamma$ alters the weight of the genus-dependent factor, allowing contributions from arbitrarily high genera to survive in the limit.

3. Combinatorial Identification:
   The paper establishes a bijection between the Lukasiewicz path configurations and classical combinatorial objects in free probability, such as non-crossing annular permutations and second-order non-crossing annular partitions, thereby linking the new formulas to existing literature.

Key Contributions and Results

1. Necessary and Sufficient Conditions:
   The paper provides precise criteria for the CLT in both regimes based on the asymptotic behavior of the coefficients $c_\lambda$ of $\log G_{N,\theta}(\vec{x}) = \sum c_\lambda p_\lambda(\vec{x})$:

   • Fixed-temperature: $c_\lambda \to 0$ for $\ell(\lambda) \ge 3$; one-row coefficients scale as $N^{-1}c_{(d)} \to \theta^{1-d}\kappa_d$; two-row coefficients scale as $c_{(r,s)} \to \theta^{1-r-s}\kappa_{r,s}$.
   • High-temperature: $N^{\ell(\lambda)/2}c_\lambda \to 0$ for $\ell(\lambda) \ge 3$; one-row coefficients scale as $d \cdot c_{(d)} \to \kappa^\gamma_d$; two-row coefficients scale as $N(1 + 1_{r=s})rs \cdot c_{(r,s)} \to \kappa^\gamma_{r,s}$.
     Here, $\kappa$ and $\kappa^\gamma$ represent first-order free cumulants and second-order free (or $\gamma$-) cumulants, respectively.

2. Explicit Covariance Formulas:
   The limiting covariance is expressed explicitly in terms of these cumulants via sums over Lukasiewicz paths:

   • Fixed-temperature: The covariance is a sum of two parts: a term involving products of first-order cumulants weighted by path statistics ($A(E)B(E)C(E)$), and a term involving second-order cumulants. The formula recovers the classical second-order free moment–cumulant relation (Collins–Mingo–Śniady–Speicher) up to a factor of $1/\theta$.
   • High-temperature: A new formula is derived involving second-order free $\gamma$-cumulants. The covariance is a sum over configurations weighted by powers of $\gamma$ and the number of down-steps in the path.

3. Topological Interpretation:
   The work reveals a distinct topological structure for the two regimes:

   • At fixed temperature, the limit is governed strictly by genus zero (planar) contributions.
   • At high temperature, the limit incorporates contributions from arbitrarily high genera, providing a topological interpretation for the new $\gamma$-cumulants.

4. Applications:

   • $\beta$-Addition: The paper proves that $\beta$-addition (a binary operation on $N$-particle systems defined by the product of BGFs) preserves the CLT in both regimes. The limiting cumulants of the sum are the sums of the input cumulants.
   • Specific Ensembles: The theory is applied to Gaussian and Laguerre $\beta$-ensembles, yielding explicit high-temperature CLTs and limiting covariances.
   • Transition: A rescaled transition from the high-temperature covariance to the fixed-temperature counterpart is established as $\gamma \to \infty$.

Significance
The paper claims to unify the fluctuation theory of random matrix ensembles across temperature regimes by moving beyond specific log-gas densities to a general framework based on Bessel generating functions.

• It extends the concept of second-order freeness to the high-temperature regime, introducing second-order free $\gamma$-cumulants which linearize $\gamma$-convolution.
• It provides a combinatorial description of the covariance that does not rely on annular non-crossing partitions but rather on Lukasiewicz paths, offering a different perspective that naturally accommodates the high-temperature scaling.
• The topological expansion offers a unified view where the distinction between fixed and high temperature is a matter of which genera dominate the asymptotic expansion, bridging the gap between classical matrix models and general continuous particle systems.

The results are presented as rigorous characterizations of the CLT, establishing the equivalence between the convergence of specific BGF coefficients and the Gaussian fluctuation of global observables, without assuming the particles are eigenvalues of a specific random matrix model.