Asymptotic analysis of a random walk on a hypercube with many dimensions

Abstract

In nearest neighbor random walk on an n‐dimensional cube a particle moves to one of its nearest neighbors (or stays fixed) with equal probability. the particle starts at 0. How long does it take to reach its stationary distribution? in fact, this occurs surprisingly rapidly. Previous analysis has shown that the total variation distance to stationarity is large if the number of steps N is < 1/4 n log n and close to 0 if N > 1/4 n log n. This paper derives an explicit expression for the variation distance as n → ∞ in the transition region N ˜ 1/4 n log n. This permits the first careful evaluation of a cutoff phenomenon observed in a wide variety of Markov chains. the argument involves Fourier analysis to express the probability as a contour integral and saddle point approximation. the asymptotic results are in good agreement with numerical results for n as small as 100.