Expanderizing Higher Order Random Walks
Vedat Levi Alev, Shravas Rao
TL;DR
This work introduces expanderized higher-order random walks on $n$-partite simplicial complexes, replacing dense update steps with sparse expander-based transitions to generalize down-up and systematic-scan dynamics. It develops a transfer framework showing that log-Sobolev and Poincaré inequalities for the base walks carry over to the expanderized variants with a degradation controlled by $\mathrm{gap}(H)$ and $\mathrm{gap}(H^2)$, and it provides entropy-contraction and local-to-global $\Phi$-entropy results via Garland-type arguments. The authors prove that expanderized up-down/down-up walks can be closely approximated by damped versions of the standard walks and derive spectral-gap and entropy-contraction bounds that enable rapid mixing in important sampling problems. They apply these results to sampling list colorings of bounded-degree graphs and Ising models with PSD interactions, achieving $O(n \log n)$ mixing times while using significantly fewer random bits than traditional Glauber dynamics. Overall, the paper presents a versatile derandomization-friendly framework that leverages expander graphs to sparsify higher-order Markov chains without sacrificing fast convergence, with broad implications for sampling in combinatorial and statistical physics models.
Abstract
We study a variant of the down-up and up-down walks over an $n$-partite simplicial complex, which we call expanderized higher order random walks -- where the sequence of updated coordinates correspond to the sequence of vertices visited by a random walk over an auxiliary expander graph $H$. When $H$ is the clique, this random walk reduces to the usual down-up walk and when $H$ is the directed cycle, this random walk reduces to the well-known systematic scan Glauber dynamics. We show that whenever the usual higher order random walks satisfy a log-Sobolev inequality or a Poincaré inequality, the expanderized walks satisfy the same inequalities with a loss of quality related to the two-sided expansion of the auxillary graph $H$. Our construction can be thought as a higher order random walk generalization of the derandomized squaring algorithm of Rozenman and Vadhan. We show that when initiated with an expander graph our expanderized random walks have mixing time $O(n \log n)$ for sampling a uniformly random list colorings of a graph $G$ of maximum degree $Δ= O(1)$ where each vertex has at least $(11/6 - ε) Δ$ and at most $O(Δ)$ colors and $O\left( \frac{n \log n}{(1 - \| J\|)^2}\right)$ for sampling the Ising model with a PSD interaction matrix $J \in R^{n \times n}$ satisfying $\| J \| \le 1$ and the external field $h \in R^n$-- here the $O(\bullet)$ notation hides a constant that depends linearly on the largest entry of $h$. As expander graphs can be very sparse, this decreases the amount of randomness required to simulate the down-up walks by a logarithmic factor. We also prove some simple results which enable us to argue about log-Sobolev constants of higher order random walks and provide a simple and self-contained analysis of local-to-global $Φ$-entropy contraction in simplicial complexes -- giving simpler proofs for many pre-existing results.