math.PR — Probability

Stationary diffusion on nonnegatively curved graphs

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

Lazy random walk on a bounded-degree graph of nonnegative Ollivier–Ricci curvature has diffusive second moment and Gaussian upper tails when started in stationarity. If the maximum degree is $D$, then $\mathbb E d(X_0,X_n)^2\le4Dn$, with optimal order of dependence on $D$. The results also hold for stationary random rooted infinite graphs. The proof combines known heat-kernel total-variation decay with a reversible energy identity for the conditional mean displacement. Its exponential form controls variation between starting vertices; martingale concentration then controls the walk itself.