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Updated: Feb 28, 2026

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Mixing of fast random walks on dynamic random permutations.

Luca Avena1, Remco van der Hofstad2, Frank den Hollander3

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Probability Theory and Related Fields
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This study analyzes random walks on dynamic random permutations. The total variation distance to uniform distribution shows a sharp, random-time jump, then deterministically decreases to zero.

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Area of Science:

  • Probability theory
  • Combinatorics
  • Statistical mechanics

Background:

  • Random walks are fundamental in modeling stochastic processes.
  • Permutations and their dynamics are key in understanding complex systems.
  • Mixing time analysis is crucial for algorithm efficiency and statistical inference.

Purpose of the Study:

  • To analyze the mixing profile of a random walk on a dynamic random permutation.
  • To investigate the impact of different permutation dynamics (coagulation-only vs. coagulation-fragmentation) on mixing.
  • To characterize the convergence to the uniform distribution in a specific dynamic regime.

Main Methods:

  • Analysis of random walks on dynamic random permutations.
  • Focus on the regime where walk evolution is much faster than permutation dynamics.
  • Mathematical analysis of total variation distance and convergence to a limit process.

Main Results:

  • For both dynamics, scaled time leads to convergence to a limit process.
  • A single jump in total variation distance occurs at a random time.
  • The distance then deterministically decreases from a function of jump time to zero.

Conclusions:

  • The mixing behavior is characterized by a sharp, random-time cutoff.
  • The post-jump decay rate is a deterministic function of the jump time.
  • Results provide insights into the mixing properties of random walks on evolving structures.