One-arm exponents of the high-dimensional Ising model

Abstract

We study the probability that the origin is connected to the boundary of the box of size n (the one-arm probability) in several percolation models related to the Ising model. We prove that different universality classes emerge at criticality: For the FK-Ising measure in a box of size n with wired boundary conditions, we prove that this probability decays as 1/n in dimensions d > 4, and as 1/n1+o(1) when d = 4. For the infinite volume FK-Ising measure, we prove that this probability decays as 1/n2 in dimensions d > 6, and as 1/n2+o(1) when d = 6. For the sourceless double random current measure, we prove that this probability decays as 1/nd−2 in dimensions d > 4, and as 1/n2+o(1) when d = 4. Additionally, for the infinite volume FK-Ising measure, we show that the one-arm probability is 1/n1+o(1) in dimension d = 4, and at least 1/n3/2 in dimension d = 5. This establishes that the FK-Ising model has upper-critical dimension equal to 6, in contrast to the Ising model, where it is known to be less or equal to 4, thus solving a conjecture of Chayes, Coniglio, Machta, and Shtengel.