Exact Solution for a Class of Random Walk on the Hypercube

Abstract

A class of families of Markov chains defined on the vertices of the n-dimensional hypercube, Ω _n ={0,1}ⁿ, is studied. The single-step transition probabilities P _n,ij , with i,j∈Ω _n , are given by $P_{n,ij}=\frac{(1-{\alpha})^{d_{ij}}}{(2-{\alpha})^{n}}$, where α∈(0,1) and d _ij is the Hamming distance between i and j. This corresponds to flip independently each component of the vertex with probability $\frac{1-{\alpha}}{2-{\alpha}}$. The m-step transition matrix $P_{n,ij}^{m}$ is explicitly computed in a close form. The class is proved to exhibit cutoff. A model-independent result about the vanishing of the first m terms of the expansion in α of $P_{n,ij}^{m}$ is also proved.