Fast Mixing Random Walks and Regularity of Incompressible Vector Fields

TL;DR

This paper establishes conditions for rapid mixing of the BallWalk algorithm in certain geometric spaces, linking mixing properties to the regularity of solutions to Laplace's equation under smooth incompressible flows.

Contribution

It introduces a novel connection between the regularity of incompressible flows and the mixing efficiency of random walks in geometric spaces.

Findings

Fast mixing occurs under smooth incompressible flows transforming convex spaces.

Regularity of Laplace's equation solutions is key to constructing such flows.

Provides sufficient conditions for rapid mixing in bounded domains.

Abstract

We show sufficient conditions under which the \textsc{BallWalk} algorithm mixes fast in a bounded connected subset of $\Real^n$. In particular, we show fast mixing if the space is the transformation of a convex space under a smooth incompressible flow. Construction of such smooth flows is in turn reduced to the study of the regularity of the solution of the Dirichlet problem for Laplace's equation.