Spectral gap and cutoff phenomenon for the Gibbs sampler of ∇φ interfaces with convex potential

2020/07/20 by Pietro Caputo, Cyril Labbé, Caputo, Pietro +3
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR

paper · pdf · doi:10.48550/arxiv.2007.10108

38 pages

arxiv created 2020/07/20 · openalex publication_date 2020/07/20 · arxiv updated 2020/07/21 · openalex created_date 2022/07/26 · openalex updated_date 2026/08/01

Abstract

We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on ℝ^N describing ∇φ interfaces with convex potentials. Under minimal assumptions on the potential, we find that the spectral gap of the process is always given by gap_N=1-cos(π/N), and that for all ε∈(0,1), its ε-mixing time satisfies T_N(ε)∼ (log N)/(2gap_N) as N→∞, thus establishing the cutoff phenomenon. The results reveal a universal behavior in that they do not depend on the choice of the potential.

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