Mixing time of random walk on dynamical random cluster

Andrea Lelli; Alexandre Stauffer

doi:10.1007/s00440-024-01262-8

Mixing time of random walk on dynamical random cluster

Andrea Lelli, Alexandre Stauffer

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Abstract

We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate μ between open and closed, following a Glauber dynamics for the random cluster model with parameters p, q. At the same time, the walker jumps at rate 1 as a simple random walk on the torus, but is only allowed to traverse open edges. We show that for small enough p the mixing time of the random walker is of order n ²/μ. In our proof we construct a non-Markovian coupling through a multi-scale analysis of the environment, which we believe could be more widely applicable.

Original language	English
Pages (from-to)	981-1043
Number of pages	63
Journal	Probability Theory and Related Fields
Volume	189
Issue number	3-4
Early online date	28 Feb 2024
DOIs	https://doi.org/10.1007/s00440-024-01262-8
Publication status	Published - 31 Aug 2024

Funding

Andrea Lelli: Most of this work was done while the author was affiliated with the University of Bath, Department of Mathematical Sciences, supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/L015684/1. Alexandre Stauffer: supported by EPSRC Fellowship EP/N004566/1.

Funders	Funder number
EPSRC Centre for Doctoral Training in Statistical	EP/L015684/1
Engineering and Physical Sciences Research Council	EP/N004566/1
Engineering and Physical Sciences Research Council

Keywords

Mixing time
Primary 60K35
Random cluster
Random walk
Secondary 60K37
Time inhomogeneous Markov chains

ASJC Scopus subject areas

Analysis
Statistics and Probability
Statistics, Probability and Uncertainty

Access to Document

10.1007/s00440-024-01262-8Licence: CC BY

2209.03227v1Submitted manuscript, 985 KBLicence: CC BY

https://arxiv.org/abs/2209.03227Licence: CC BY

Cite this

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title = "Mixing time of random walk on dynamical random cluster",

abstract = "We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate μ between open and closed, following a Glauber dynamics for the random cluster model with parameters p, q. At the same time, the walker jumps at rate 1 as a simple random walk on the torus, but is only allowed to traverse open edges. We show that for small enough p the mixing time of the random walker is of order n 2/μ. In our proof we construct a non-Markovian coupling through a multi-scale analysis of the environment, which we believe could be more widely applicable.",

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N2 - We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate μ between open and closed, following a Glauber dynamics for the random cluster model with parameters p, q. At the same time, the walker jumps at rate 1 as a simple random walk on the torus, but is only allowed to traverse open edges. We show that for small enough p the mixing time of the random walker is of order n 2/μ. In our proof we construct a non-Markovian coupling through a multi-scale analysis of the environment, which we believe could be more widely applicable.

AB - We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate μ between open and closed, following a Glauber dynamics for the random cluster model with parameters p, q. At the same time, the walker jumps at rate 1 as a simple random walk on the torus, but is only allowed to traverse open edges. We show that for small enough p the mixing time of the random walker is of order n 2/μ. In our proof we construct a non-Markovian coupling through a multi-scale analysis of the environment, which we believe could be more widely applicable.

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KW - Primary 60K35

KW - Random cluster

KW - Random walk

KW - Secondary 60K37

KW - Time inhomogeneous Markov chains

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JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

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ER -

Mixing time of random walk on dynamical random cluster

Abstract

Funding

Keywords

ASJC Scopus subject areas

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Early Career Fellowship - Mathematical Analysis of Strongly Correlated Processes on Discrete Dynamic Structures

Cite this

Mixing time of random walk on dynamical random cluster

Abstract

Funding

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Projects

Early Career Fellowship - Mathematical Analysis of Strongly Correlated Processes on Discrete Dynamic Structures

Cite this