For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (Comm Math Phys 270(2):335–358, 2007). This improvement finally settles a conjecture by Aizenman (Nuclear Phys B 485(3):551–582, 1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (Ann Probab 36(4):1267–1286, 2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. Finally, we show that any weak limit of the largest connected component is non-degenerate, which can be viewed as a significant sign of critical behavior. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdős-Rényi random graphs.
Percolation Random graph asymptotics Mean-field behavior Critical window
The work of RvdH was supported in part by the Netherlands Organisation for Scientific Research (NWO). We thank Asaf Nachmias for enlightening discussions concerning the results and methodology in [16,19]. MH is grateful to Institut Mittag-Leffler for the kind hospitality during his stay in February 2009, and in particular to Jeff Steif for inspiring discussions.
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Borgs C., Chayes J.T., van der Hofstad R., Slade G., Spencer J.: Random subgraphs of finite graphs. I. The scaling window under the triangle condition. Random Struct. Algorithms 27(2), 137–184 (2005)zbMATHCrossRefGoogle Scholar
Borgs C., Chayes J.T., van der Hofstad R., Slade G., Spencer J.: Random subgraphs of finite graphs. II. The lace expansion and the triangle condition. Ann. Probab. 33(5), 1886–1944 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
Grimmett G.: Percolation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, 2nd edn. Springer, Berlin (1999)Google Scholar
Hara T., van der Hofstad R., Slade G.: Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31(1), 349–408 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
Hara T., Slade G.: The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents. J. Stat. Phys. 99(5–6), 1075–1168 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
Hara T., Slade G.: The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. J. Math. Phys. 41(3), 1244–1293 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.