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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
1Dipartimento di Matematica e Informatica 'Ulisse Dini', Università degli Studi di Firenze, Florence, Italy.
Probability Theory and Related Fields
|February 26, 2026
Summary
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.
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.
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