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Random graph asymptotics on high-dimensional tori. II: volume, diameter and mixing time. (English) Zbl 1229.60108
Probab. Theory Relat. Fields 149, No. 3-4, 397-415 (2011); correction ibid. 175, No. 3-4, 1183-1185 (2019).
The authors study critical bond-percolation on high-dimensional tori, removing a logarithmic correction in the lower bound considered in their previous paper [M. Heydenreich and R. van der Hofstad, Commun. Math. Phys. 270, No. 2, 335–358 (2007; Zbl 1128.82010)] and prove sharp lower bounds on the size of the largest cluster. The main result of the paper is that the behavior of the critical percolation on high-dimensional tori is the same as for critical Erdös-Rényi random graphs.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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