Power-law bounds for critical long-range percolation below the upper-critical dimension

• Published in: Probability theory and related fields 2021-11, Vol.181 (1-3), p.533-570
• Main Author: Hutchcroft, Tom
• Format: Article
• Language: English
• Subjects: Apexes; Clusters; Economics; Finance; Graph theory; Inequality; Insurance; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Percolation; Power law; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Statistics for Business; Theorems; Theoretical; Upper bounds
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-021-01043-7

We study long-range Bernoulli percolation on Z d in which each two vertices x and y are connected by an edge with probability 1 - exp ( - β ‖ x - y ‖ - d - α ) . It is a theorem of Noam Berger ( Commun. Math. Phys. , 2002) that if 0 < α < d then there is no infinite cluster at the critical parameter β c . We give a new, quantitative proof of this theorem establishing the power-law upper bound P β c ( | K | ≥ n ) ≤ C n - ( d - α ) / ( 2 d + α ) for every n ≥ 1 , where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality ( 2 - η ) ( δ + 1 ) ≤ d ( δ - 1 ) relating the cluster-volume exponent δ and two-point function exponent η .