Small-ball estimates for random walks on groups

We prove a new inequality bounding the probability that the random walk on a group has small total displacement in terms of the spectral and isoperimetric profiles of the group. This inequality implies that if the random walk on the group is diffusive then Cheeger's inequality is sharp in the sense that the isoperimetric profile $\Phi$ and spectral profile $\Lambda$ of the group are related by $\Lambda \simeq \Phi^2$. Our inequality also yields substantial progress on a conjecture of Lyons, Peres, Sun, and Zheng (2017) stating that for any transient random walk on an infinite, finitely generated group, the expected occupation time of the ball of radius $r$ is $O(r^2)$: We prove that this conjecture holds for every group of superpolynomial growth whose spectral profile is slowly varying, which we conjecture is always the case. For groups of exponential or stretched-exponential growth satisfying a further mild regularity assumption on their spectral profile, our method yields the strong quantitative small-ball estimate $$-\log \mathbb{P}\bigl(d(X_0,X_n) \leq \varepsilon n^{1/2}\bigr) \succeq \frac{1}{\varepsilon^2} \wedge (-\log \mathbb{P}(X_n=X_0)),$$ which is sharp for the lamplighter group. Finally, we prove that the regularity assumptions needed to apply the strongest versions our results are satisfied for several classical examples where the spectral profile is not known explicitly, including the first Grigorchuk group and Thompson's group $F$.