Abstract
The cutoff phenomenon describes a sharp transition in the convergence of an ergodic finite Markov chain to equilibrium. Of particular interest is understanding this convergence for the simple random walk on a bounded-degree expander graph. The first example of a family of bounded-degree graphs where the random walk exhibits cutoff in total-variation was provided only very recently, when the authors showed this for a typical random regular graph. However, no example was known for an explicit (deterministic) family of expanders with this phenomenon. Here we construct a family of cubic expanders where the random walk from a worst case initial position exhibits total-variation cutoff. Variants of this construction give cubic expanders without cutoff, as well as cubic graphs with cutoff at any prescribed time-point.
Highlights
A finite ergodic Markov chain is said to exhibit cutoff in total-variation if its L1-distance from the stationary distribution drops abruptly from near its maximum to near 0
Note that this condition clearly holds for the simple random walk on an n-vertex expander, where the inverse-gap is O(1) whereas tmix log n
The first example of a family of bounded-degree graphs where the random walk exhibits cutoff in total-variation was provided only very recently [15], when the authors showed this for a typical random regular graph
Summary
A finite ergodic Markov chain is said to exhibit cutoff in total-variation if its L1-distance from the stationary distribution drops abruptly from near its maximum to near 0. In 2004, Peres[16] observed that for any family of reversible Markov chains, total-variation cutoff can only occur if the inverse spectral-gap has smaller order than the mixing time Note that this condition clearly holds for the simple random walk on an n-vertex expander, where the inverse-gap is O(1) whereas tmix log n. It is well known that for any fixed d ≥ 3, a random d-regular graph is with high probability (w.h.p.) a very good expander, the simple random walk on almost every d-regular expander exhibits worst-case total-variation cutoff To this date there were no known examples for an explicit (deterministic) family of expanders with this phenomenon. For any family of bounded-degree n-vertex graphs where the SRW has tmix n2 (largest possible order of mixing) there cannot be cutoff
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