A recurring theme in statistical learning, online learning, and beyond is that faster
convergence rates are possible for problems with low noise, often quantified by the …

We prove Schramm's locality conjecture for Bernoulli bond percolation on transitive graphs:
If $(G_n) _ {n\geq 1} $ is a sequence of infinite vertex-transitive graphs converging locally to …

We study harmonic functions on random environments with particular emphasis on the case
of the infinite cluster of supercritical percolation on Z^d. We prove that the vector space of …

The fundamental inequality of Guivarc'h relates the entropy and the drift of random walks on
groups. It is strict if and only if the random walk does not behave like the uniform measure on …

We present a survey of results related to Milnor's problem on group growth. We discuss the
cases of polynomial growth and exponential but not uniformly exponential growth; the main …

Abstract We study the Poisson–Furstenberg boundary of random walks on permutational
wreath products. We give a sufficient condition for a group to admit a symmetric measure of …

We prove an essentially sharp lower bound on the k-round distributional complexity of the k-
step pointer chasing problem under the uniform distribution, when Bob speaks first. This is …

Let G be a finitely generated group equipped with a finite symmetric generating set and the
associated word length function |⋅|. We study the behavior of the probability of return for …

Research in recent years has highlighted the deep connections between the algebraic,
geometric, and analytic structures of a discrete group. New methods and ideas have …

Th e entropy, the spectral radius and the drift are important numerical quantities associated
to random walks on countable groups. We prove sharp inequalities relating those quantities …