Bounds for the annealed return probability on large finite percolation clusters
Probability
2017-06-20 v5 Combinatorics
Abstract
Bounds for the expected return probability of the delayed random walk on finite clusters of an invariant percolation on transitive unimodular graphs are derived. They are particularly suited for the case of critical Bernoulli percolation and the associated heavy-tailed cluster size distributions. The upper bound relies on the fact that cartesian products of finite graphs with cycles of a certain minimal size are Hamiltonian. For critical Bernoulli bond percolation on the homogeneous tree this bound is sharp. The asymptotic type of the expected return probability for large times t in this case is of order of the 3/4'th power of 1/t.
Cite
@article{arxiv.0812.0117,
title = {Bounds for the annealed return probability on large finite percolation clusters},
author = {Florian Sobieczky},
journal= {arXiv preprint arXiv:0812.0117},
year = {2017}
}
Comments
New result for the particular case of homogeneous trees illustrates sharpness of the bound