2022/06/16 by Florentin Münch, Justin Salez, Münch, Florentin +1 · 2 citations
Mathematics · Environmental Science · #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Per- and polyfluoroalkyl substances research
paper · pdf · doi:10.48550/arxiv.2206.08294
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.