The Liouville property and Hilbertian compression

Lower bound on the equivariant Hilbertian compression exponent $\alpha$ are obtained using random walks. More precisely, if the probability of return of the simple random walk is $\succeq \textrm{exp}(-n^\gamma)$ in a Cayley graph then $\alpha \geq (1-\gamma)/(1+\gamma)$. This motivates the study of further relations between return probability, speed, entropy and volume growth. For example, if $|B_n| \preceq e^{n^\nu}$ then the speed is $\preceq n^{1/(2-\nu)}$. Under a strong assumption on the off-diagonal decay of the heat kernel, the lower bound on compression improves to $\alpha \geq 1-\gamma$. 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 $\gamma <1/2$.