Spectral Independence A New Tool to Analyze Markov Chains

Abstract

We introduce a versatile technique called spectral independence for the analysis of Markov chainMonte Carlo algorithms in high-dimensional probability and statistics. We rigorously prove rapid mixing of practically usefully Markov chains for sampling from important classes of probability distributions arising in computer science, statistical physics, and pure mathematics, thus resolving several longstanding conjectures and open problems. In many cases, we obtain asymptotically optimal mixing time bounds. To achieve these results, we establish new local-to-global phenomena which translate spectral independence into mixing time bounds. Furthermore, we develop four distinct classes of techniques for establishing spectral independence by building new bridges with other fields.