Limit Profiles for Reversible Markov Chains

Abstract

In a recent breakthrough, Teyssier [Tey20] introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the $k$-cycle shuffle, improving results of Hough [Hou16] and Berestycki, Schramm and Zeitouni [BSZ11]; the Ehrenfest urn diffusion with many urns, improving results of Ceccherini-Silberstein, Scarabotti and Tolli [CST07]; a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, improving results of Diaconis, Khare and Saloff-Coste [DKS08].

v3. Minor mistake corrected in an error term in proof of Theorem B/3.1. Cutoff window shifted accordingly

References in corpus (4)

• Mixing times for random k-cycles and coalescence-fragmentation chains
• Limit Profiles for Reversible Markov Chains
• Cutoff for Almost All Random Walks on Abelian Groups
• Gibbs Sampling, Exponential Families and Orthogonal Polynomials

Cited by in corpus (9)

• Limit Profiles for Reversible Markov Chains
• Cutoff profile of the Metropolis biased card shuffling
• On the spectrum and ergodicity of a neutral multi-allelic Moran model
• Limit Profile for Projections of Random Walks on Groups
• Cutoff for Rewiring Dynamics on Perfect Matchings
• Gradual convergence for Langevin dynamics on a degenerate potential
• The cutoff phenomenon for the stochastic heat and the wave equation subject to small Lévy noise
• Cutoff phenomenon for the warp-transpose top with random shuffle
• Limit profile for the transpose top-2 with random shuffle