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Random graph asymptotics on high-dimensional tori. (English) Zbl 1128.82010
In the paper it is considered Bernoulli bond percolation on the hypercubic lattice \(\mathbb Z^d\) and the finite torus \(T_{r,d} = \{-[r/2], \dots, [r/2]- 1\}^d\). The scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in sufficiently high dimensions, or when \(d > 6\) for sufficiently spread-out percolation is investigated. Using coupling argument it is established that the largest critical cluster is, with high probability, bounded above by a large constant times \(V^{2/3}\) and below by a small constant times \(V^{2/3}(\log V)^{-4/3}\), where \(V\) is the volume of the torus. A simple criterion in terms of the subcritical percolation two-point function on \(\mathbb Z^d\) is given under which the lower bound can be improved to small constant times \(V^{2/3}\), i.e., it is proven random graph asymptotics for the largest critical cluster on the high-dimensional torus.
This establishes a conjecture by M. Aizenman [Nucl. Phys., B 485, No. 3, 551–582 (1997; Zbl 0925.82112)], apart from logarithmic corrections. Other implications of these results on the dependence on boundary conditions for high-dimensional percolation are disscused. The used method is crucially based on the results of [C. Borgs, J. T. Chayes, R. van der Hofstad, G. Slade and J. Spencer, Ann. Probab. 33, No. 5, 1886–1944 (2005; Zbl 1079.05087); R. van der Hofstad and G. Slade, Random Struct. Algorithms 27, No. 3, 331–357 (2005; Zbl 1077.60077)], where the \(V^{2/3}\) scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on \(\mathbb Z^d\).

82B43 Percolation
60K35 Interacting random processes; statistical mechanics type models; percolation theory
05C80 Random graphs (graph-theoretic aspects)
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