Rapid Mixing of [math]-Class Biased Permutations

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 702-725, March 2024.
Abstract. In this paper, we study a biased version of the nearest-neighbor transposition Markov chain on the set of permutations where neighboring elements [math] and [math] are placed in order [math] with probability [math]. Our goal is to identify the class of parameter sets [math] for which this Markov chain is rapidly mixing. Specifically, we consider the open conjecture of Jim Fill [Background on the Gap Problem (2003) and An Interesting Spectral Gap Problem (2003)] that all monotone, positively biased distributions are rapidly mixing. We resolve Fill’s conjecture in the affirmative for distributions arising from [math]-class particle processes, where the elements are divided into [math] classes and the probability of exchanging neighboring elements depends on the particular classes the elements are in. We further require that [math] is a constant and that all probabilities between elements in different classes are bounded away from [math]. These particle processes arise in the context of self-organizing lists, and our result also applies beyond permutations to the setting where all particles in a class are indistinguishable. Our work generalizes recent work by Haddadan and Winkler [Mixing of permutations by biased transposition (2017)] studying 3-class particle processes. Additionally, we show that a broader class of distributions based on trees is also rapidly mixing, which generalizes a class analyzed by Bhakta et al. [Mixing times of Markov chains for self-organizing lists and biased permutations (2013)]. Our proof involves analyzing a generalized biased exclusion process, which is a nearest-neighbor transposition chain applied to a 2-particle system. Biased exclusion processes are of independent interest, with applications in self-assembly. We generalize the results of Greenberg et al. [Sampling biased lattice configurations using exponential metrics (2009)] and Benjamini et al. [Mixing times of the biased card shuffling and the asymmetric exclusion process (2005)] on biased exclusion processes to allow the probability of swapping neighboring elements to depend on the entire system, as long as the minimum bias is bounded away from 1.