The Liouville property and Hilbertian compression

Abstract

Lower bound on the equivariant Hilbertian compression exponent $α$ are obtained using random walks. More precisely, if the probability of return of the simple random walk is $\succeq \textrm{exp}(-n^γ)$ in a Cayley graph then $α\geq (1-γ)/(1+γ)$. This motivates the study of further relations between return probability, speed, entropy and volume growth. For example, if $|B_n| \preceq e^{n^ν}$ then the speed is $\preceq n^{1/(2-ν)}$. Under a strong assumption on the off-diagonal decay of the heat kernel, the lower bound on compression improves to $α\geq 1-γ$. Using a result from Naor and Peres on compression and the speed of random walks, this yields very promising bounds on speed and implies the Liouville property if $γ<1/2$.