Bounds for the annealed return probability on large finite random percolation clusters

Details Bounds for the annealed return probability on large finite random percolation clusters Bounds for the annealed return probability on large finite random percolation clusters

2008-11-30

arxiv.org/abs/0812.0117v4

Mathematics

Probability

New lower bound (planar graphs); added references

Scientific paper

By an eigenvalue comparison-technique polynomial bounds for the expected return probability of the delayed random walk on critical Bernoulli bond percolation clusters are derived. The results refer to invariant percolations on unimodular transitive planar graphs with almost surely finite critical clusters. Estimates for the integrated density of states of the graph Laplacian of the two-dimensional Euclidean lattice follow. The upper bound which also applies to non-planar graphs relies on the fact that Cartesian products of finite graphs with cycles of a certain minimal size are Hamiltonian. The lower bound involves an upper estimate of the isoperimetric number (`Cheeger-constant') of finite graphs.

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