## Electronic Journal of Probability

### Bounds for the annealed return probability on large finite percolation graphs

Florian Sobieczky

#### 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 $t^{-3/4}$.

#### Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 79, 17 pp.

Dates
Accepted: 21 September 2012
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465062401

Digital Object Identifier
doi:10.1214/EJP.v17-2329

Mathematical Reviews number (MathSciNet)
MR2981904

Zentralblatt MATH identifier
1260.05149

Rights

#### Citation

Sobieczky, Florian. Bounds for the annealed return probability on large finite percolation graphs. Electron. J. Probab. 17 (2012), paper no. 79, 17 pp. doi:10.1214/EJP.v17-2329. https://projecteuclid.org/euclid.ejp/1465062401

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