Restricted Percolation Critical Exponents in High Dimensions

Despite great progress in the study of critical percolation onDOUBLE-STRUCK CAPITAL Z(d)fordlarge, 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 that0and(n,n,n, horizontal ellipsis )are connected within[-n,n](d)scales asn(-2 - 2d). 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") from0to the boundary of a sidelengthnbox scales asn(-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. (c) 2020 Wiley Periodicals LLC