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A note about critical percolation on finite graphs. (English) Zbl 1235.60138
Summary: We study the geometry of the largest component $${\mathcal{C}}_{1}$$ of the critical percolation on a finite graph $$G$$ which satisfies the finite triangle condition, defined by C. Borgs et al. [Random Struct. Algorithms 27, No. 2, 137–184 (2005; Zbl 1076.05071)]. There it was 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. In the light of [C. Borgs et al., Ann. Probab. 33, No. 5, 1886–1944 (2005; Zbl 1079.05087)], our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus $${\mathbb{Z}}_{n}^{d}$$ (with $$d$$ large and $$n \rightarrow \infty$$) and the Hamming cube $$\{0,1\}^{n}$$.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 05C80 Random graphs (graph-theoretic aspects)
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