Florian Sobieczky,
Mathematisches Institut Jena, Universität Jena

Annealed bounds for the return probability of the simple random walk on
critical percolation clusters


Abstract

Critical Bernoulli percolation on a unimodular transitive graph and on the 2-dim. 
euclidean lattice has almost surely finite connected components. Estimating the 
expected return probability of the simple random walk is  difficult, due to the 
heavy tails of the cluster-size distribution. Annealed upper and lower bounds are 
presented and compared to the conventional technique: instead of only estimating 
the spectral gap, the whole spectrum of the graph Laplacian is involved. In the case 
of regular trees, this is good enough to distinguish the decay of the expected return 
probability on finite clusters from that on the incipient infinite cluster.