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Normal typicality and dynamical typicality for a random block-band matrix model.

László Erdős1, Joscha Henheik2, Cornelia Vogel3

  • 1Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria.

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|December 29, 2025
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Summary
This summary is machine-generated.

This study proves normal and dynamical typicality for random block-band matrices. It introduces a novel model demonstrating intermediate equilibration times, a significant advancement in random matrix theory.

Keywords:
Dynamical typicalityEquilibrationNormal typicalityQuantum dynamicsWigner type matrix

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Area of Science:

  • Mathematics
  • Probability Theory
  • Random Matrix Theory

Background:

  • Random matrix theory analyzes properties of large random matrices.
  • Typicality in random matrices describes the convergence of matrix properties to deterministic limits.
  • Equilibration times measure how quickly a system reaches a steady state.

Purpose of the Study:

  • To rigorously prove normal typicality and dynamical typicality for a specific random block-band matrix model.
  • To establish intermediate equilibration times for this model, a previously unproven aspect.
  • To advance the understanding of random matrix behavior and their dynamic properties.

Main Methods:

  • Development of a centered random block-band matrix model with block-dependent variances.
  • Application of recently established concentration estimates for products of resolvents of Wigner type random matrices.
  • Intricate analysis of the deterministic approximation to bridge the gap between random and deterministic behaviors.

Main Results:

  • Successful proof of normal typicality for the random block-band matrix model.
  • Successful proof of dynamical typicality for the random block-band matrix model.
  • Demonstration of intermediate equilibration times, a novel rigorous result in this field.

Conclusions:

  • The study provides a rigorous framework for understanding typicality in complex random matrix models.
  • The findings on intermediate equilibration times offer new insights into the dynamics of random systems.
  • This work contributes significantly to the theoretical foundations of random matrix theory and its applications.