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 ⪰exp(−nγ) in a Cayley graph
then α≥(1−γ)/(1+γ). This motivates the study of further
relations between return probability, speed, entropy and volume growth. For
example, if ∣Bn∣⪯enν then the speed is ⪯n1/(2−ν).
Under a strong assumption on the off-diagonal decay of the heat kernel, the
lower bound on compression improves to α≥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.Comment: 16 page