Random Graph Asymptotics on High-Dimensional Tori

• Published in: Communications in mathematical physics Vol. 270; no. 2; pp. 335 - 358
• Main Authors: Heydenreich, Markus, van der Hofstad, Remco
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01.03.2007; Springer Nature B.V
• Subjects: Asymptotic properties; Boundary conditions; Clusters; Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Graph theory; Lower bounds; Mathematical methods in physics; Mathematics; Percolation; Physics; Probability and statistics; Probability theory and stochastic processes; Probability theory, stochastic processes, and statistics; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Toruses; Lower bound; High-dimensional torus; Percolation; Probability distribution; Boundary condition; Mathematical physics; Random graph
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-006-0152-8

We investigate 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. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times V2/3 and below by a small constant times , where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on under which the lower bound can be improved to small constant times , i.e. we prove random graph asymptotics for the largest critical cluster on the high-dimensional torus. This establishes a conjecture by [1], apart from logarithmic corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation.Our method is crucially based on the results in [11, 12], where the scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on . We also strongly rely on mean-field results for percolation on proved in [17–20].