Mixing Time Bounds for Asymmetric Random Walks on Hypercubes, Cycles, and Tori Using Coupling

Abstract

Mixing time bounds for Markov chains play a central role in characterizing the sample complexity of learning and inference from correlated data. While the mixing behavior of symmetric random walks on standard graph structures such as cycles, tori, and hypercubes is well understood, the impact of transition asymmetry remains less explored. In this work, we study the mixing times of lazy, asymmetric random walks on cycles, tori, and hypercubes, motivated by their relevance in practical applications. For the $n$-cycle, we develop a novel coupling construction that yields an order-wise tight upper bound $O\left(\frac{n^{2}}{p+q}\right)$, explicitly capturing the dependence on asymmetric transition probabilities $p$ and $q$. Numerical results indicate that this dependence is highly accurate. Building on this result, we derive corresponding bounds for $d$-dimensional tori. For asymmetric random walks on $n$-dimensional hypercubes, motivated by applications, we consider the problem of estimating expectations of functions that depend only on a subset $\Delta \ll n$ of coordinates. We show that the effective sample complexity improves to $O(n \log \Delta)$, compared to $O(n \log n)$ for the full chain. In all cases, our bounds recover the tightest known results for symmetric walks as special cases.