Mixing time and expansion of non-negatively curved Markov chains

TL;DR

This paper explores how non-negative curvature influences the mixing time, conductance, and cutoff phenomena in sparse Markov chains, revealing new bounds and disproving cutoff under certain conditions.

Contribution

It establishes three key consequences of non-negative curvature on Markov chains: conductance decay, diffusive displacement until mixing, and absence of cutoff, answering longstanding questions.

Findings

Conductance decreases logarithmically with states

Displacement remains diffusive until mixing time

No cutoff phenomenon occurs in these chains

Abstract

We establish three remarkable consequences of non-negative curvature for sparse Markov chains. First, their conductance decreases logarithmically with the number of states. Second, their displacement is at least diffusive until the mixing time. Third, they never exhibit the cutoff phenomenon. The first result provides a nearly sharp quantitative answer to a classical question of Ollivier, Milman and Naor. The second settles a conjecture of Lee and Peres for graphs with non-negative curvature. The third offers a striking counterpoint to the recently established cutoff for non-negatively curved chains with uniform expansion.