  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 $\gamma
<1/2$.
