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A Note About Critical Percolation on Finite Graphs

Author

Listed:
  • Gady Kozma

    (The Weizmann Institute of Science)

  • Asaf Nachmias

    (Microsoft Research)

Abstract

In this note we study the geometry of the largest component $\mathcal {C}_{1}$ of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. (Random Struct. Algorithms 27:137–184, 2005). 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. By Borgs et al. (Ann. Probab. 33:1886–1944, 2005), 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→∞) and the Hamming cube {0,1} n .

Suggested Citation

  • Gady Kozma & Asaf Nachmias, 2011. "A Note About Critical Percolation on Finite Graphs," Journal of Theoretical Probability, Springer, vol. 24(4), pages 1087-1096, December.
  • Handle: RePEc:spr:jotpro:v:24:y:2011:i:4:d:10.1007_s10959-010-0283-x
    DOI: 10.1007/s10959-010-0283-x
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