Fast mixing of a randomized shift-register Markov chain

TL;DR

This paper introduces a Markov chain on the hypercube combining random and deterministic moves, demonstrating rapid mixing with a cutoff phenomenon and a mixing time proportional to the dimension n.

Contribution

It presents a new Markov chain with a linear shift register that achieves optimal mixing time and exhibits a sharp cutoff, advancing understanding of mixing behaviors on hypercubes.

Findings

Mixing time is approximately n, matching the dimension.

The chain exhibits a cutoff with a small window of size O(n^{0.5+δ}).

The chain combines random and deterministic moves for efficient mixing.

Abstract

We present a Markov chain on the $n$-dimensional hypercube $\{0,1\}^n$ which satisfies $t_{{\rm mix}}(\epsilon) = n[1 + o(1)]$. This Markov chain alternates between random and deterministic moves and we prove that the chain has cut-off with a window of size at most $O(n^{0.5+\delta})$ where $\delta>0$. The deterministic moves correspond to a linear shift register.