A note about critical percolation on finite graphs

Details A note about critical percolation on finite graphs A note about critical percolation on finite graphs

2009-09-24

arxiv.org/abs/0909.4351v2

Mathematics

Probability

11 pages

Scientific paper

In this note we study the geometry of the largest component C_1 of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. There it is shown that this component is of size n^{2/3}, and here we show that its diameter is n^{1/3} and that the simple random walk takes n steps to mix on it. Our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus Z_n^d (with d large and n tending to infinity) and the Hamming cube {0,1}^n.

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