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 $⪰\text{exp}(-n^{\gamma })$ in a Cayley graph then $\alpha \ge (1-\gamma )\mathrm{/}(1+\gamma )$. This motivates the study of further relations between return probability, speed, entropy and volume growth. For example, if $\mathrm{\mid }{B}_n\mathrm{\mid }⪯e^{n^{\nu }}$ 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 $\alpha \ge 1-\gamma$. 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 $\gamma <1/2$.