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Restricted Percolation Critical Exponents in High Dimensions
Author(s) -
Chatterjee Shirshendu,
Hanson Jack
Publication year - 2020
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21938
Subject(s) - mathematics , connection (principal bundle) , scaling , percolation critical exponents , random walk , critical exponent , percolation (cognitive psychology) , critical point (mathematics) , conjecture , statistical physics , percolation threshold , boundary (topology) , combinatorics , mathematical analysis , geometry , statistics , physics , quantum mechanics , neuroscience , biology , electrical resistivity and conductivity
Despite great progress in the study of critical percolation on ℤ d for d large, properties of critical clusters in high‐dimensional fractional spaces and boxes remain poorly understood, unlike the situation in two dimensions. Closely related models such as critical branching random walk give natural conjectures for the value of the relevant high‐dimensional critical exponents; see in particular the conjecture by Kozma‐Nachmias that the probability that 0 and ( n , n , n , …) are connected within [− n , n ] d scales as n −2 − 2 d . In this paper, we study the properties of critical clusters in high‐dimensional half‐spaces and boxes. In half‐spaces, we show that the probability of an open connection (“arm”) from 0 to the boundary of a sidelength n box scales as n −3 . We also find the scaling of the half‐space two‐point function (the probability of an open connection between two vertices) and the tail of the cluster size distribution. In boxes, we obtain the scaling of the two‐point function between vertices which are any macroscopic distance away from the boundary. Our argument involves a new application of the “mass transport" principle which we expect will be useful to obtain quantitative estimates for a range of other problems. © 2020 Wiley Periodicals LLC