CLT for -Ensembles at High Temperature and for Integrable Systems: A Transfer Operator Approach

Area of Science:

• Mathematical Physics
• Probability Theory
• Statistical Mechanics

Background:

• Integrable models and random matrix ensembles are crucial in understanding complex systems.
• High-temperature expansions and polynomial potentials are key areas of statistical physics.
• Central Limit Theorems (CLTs) are fundamental for statistical analysis.

Purpose of the Study:

• To prove a polynomial central limit theorem for specific integrable models and matrix ensembles.
• To establish connections between the statistical properties of integrable systems and matrix ensembles.
• To investigate the spatial correlation decay and establish Berry-Esseen bounds for these systems.

Main Methods:

• Application of advanced probability theory to integrable systems.
• Analysis of Lax matrices and their moments.
• High-temperature expansions for polynomial potentials.
• Derivation of Berry-Esseen-type bounds.

Main Results:

• A polynomial central limit theorem is proven for the studied models and ensembles.
• Mean values, variances, and correlations of Lax matrix moments are linked between integrable systems and matrix ensembles.
• Exponential decay of local functions' space-correlations is demonstrated for integrable systems.
• Berry-Esseen-type bounds are established for the considered models.

Conclusions:

• The findings provide a rigorous statistical framework for analyzing integrable systems and matrix ensembles.
• The established connections offer new insights into the relationship between different mathematical physics models.
• The results contribute to a deeper understanding of statistical properties and convergence rates in these systems.