2014/03/05 by Antoine Gournay, Gournay, Antoine
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR) #math.GR #math.PR
paper · pdf · doi:10.48550/arxiv.1403.1195
16 pages
arxiv created 2015/12/20 · arxiv updated 2015/12/22
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 \succeq \textrmexp(-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| \preceq enν then the speed is \preceq 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.