Hydrodynamic limit and cutoff for the biased adjacent walk on the simplex

2022/03/08 by Labbé, Cyril, Petit, Enguérand · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2203.04008

Abstract

We investigate the asymptotic in N of the mixing times of a Markov dynamics on N-1 ordered particles in an interval. This dynamics consists in resampling at independent Poisson times each particle according to a probability measure on the segment formed by its nearest neighbours. In the setting where the resampling probability measures are symmetric, the asymptotic of the mixing times were obtained and a cutoff phenomenon holds. In the present work, we focus on an asymmetric version of the model and we establish a cutoff phenomenon. An important part of our analysis consists in the derivation of a hydrodynamic limit, which is given by a non-linear Hamilton-Jacobi equation with degenerate boundary conditions.

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