Cycle structure of percolation on high-dimensional tori

Details Cycle structure of percolation on high-dimensional tori Cycle structure of percolation on high-dimensional tori

2011-09-06

arxiv.org/abs/1109.1233v1

Mathematics

Probability

31 pages, 2 figures

Scientific paper

In the past years, many properties of the critical behavior of the largest connected components on the high-dimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one for percolation on the complete graph or Erd\H{o}s-R\'enyi random graph, raising the question whether the scaling limits or the largest connected components, as identified by Aldous (1997), are also equal. In this paper, we investigate the cycle structure of the largest critical components for high-dimensional percolation on the torus {1,..., r}^d. While percolation clusters naturally have many short cycles, we show that the long cycles, i.e., cycles that pass through the boundary of the cube of width r/2 centered around each of their vertices, have length of order r^{d/3}, as on the critical Erd\H{o}s-R\'enyi random graph. On the Erd\H{o}s-R\'enyi random graph, cycles play an essential role in the scaling limit of the large critical clusters, as identified by Addario-Berry, Broutin and Goldschmidt arXiv:0908.3629 . Our proofs crucially rely on extensions of results of Kozma and Nachmias arXiv:0911.0871 about the one-arm exponent for critical high-dimensional percolation. We show that, at criticality, the length of an open path from the origin to (Euclidean) distance n, if it exists, is of order n^2, which is the same as for critical branching random walk.

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