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Abstract
We study the mixing time of a random walker who moves inside a dynamical random cluster model on the ddimensional torus of sidelength 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 nonMarkovian coupling through a multiscale analysis of the environment, which we believe could be more widely applicable.
Original language  English 

Pages (fromto)  9811043 
Number of pages  63 
Journal  Probability Theory and Related Fields 
Volume  189 
Issue number  34 
Early online date  28 Feb 2024 
DOIs  
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
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 1 Finished

Early Career Fellowship  Mathematical Analysis of Strongly Correlated Processes on Discrete Dynamic Structures
Stauffer, A. (PI)
Engineering and Physical Sciences Research Council
1/04/16 → 30/09/22
Project: Research council