Lower bounds on the equivariant Hilbertian compression exponent a are obtained using random walks. More precisely, if the probability of return of the simple random walk is > exp(-n(gamma)) in a Cayley graph then a >= (1-gamma)/(1+gamma). This motivates the study of further relations between return probability, speed, entropy and volume growth. For example, if vertical bar B-n vertical bar <= e(n nu) then the speed exponent is.<= 1/(2 - v).
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