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Fluctuation Moments for Regular Functions of Wigner Matrices.
1IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria.
Mathematical Physics, Analysis, and Geometry
|June 24, 2024
Summary
This study introduces a deterministic approximation for mixed fluctuation moments involving Wigner matrices and Sobolev functions. The findings extend combinatorial principles beyond free probability theory, aiding in thermalization problem analysis.
Area of Science:
- Mathematics
- Probability Theory
- Mathematical Physics
Background:
- Wigner matrices are fundamental in random matrix theory.
- Understanding fluctuations in matrix products is crucial for statistical physics and quantum mechanics.
- Existing theories like second-order free probability theory have limitations.
Purpose of the Study:
- To compute the deterministic approximation for mixed fluctuation moments of products of deterministic matrices and general Sobolev functions of Wigner matrices.
- To validate and extend combinatorial results related to non-crossing partitions and annular non-crossing permutations.
- To characterize variance in functional central limit theorems and analyze thermalization problems.
Main Methods:
- Development of deterministic approximation formulas for mixed fluctuation moments.
- Application of these formulas to products of deterministic matrices and Sobolev functions of Wigner matrices.
- Comparison with existing results for polynomial cases to confirm combinatorial validity.
Main Results:
- Formulas for deterministic approximation of mixed fluctuation moments were successfully computed.
- The study confirms the validity of non-crossing partition and annular non-crossing permutation combinatorics beyond second-order free probability.
- The derived formulas characterize variance in functional central limit theorems.
Conclusions:
- The computed deterministic approximation provides a robust tool for analyzing Wigner matrix properties.
- The results bridge random matrix theory with functional analysis and statistical physics.
- This work offers insights into fluctuations relevant to thermalization phenomena.
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