Paper
One-arm exponents of the high-dimensional Ising model
Authors
Diederik van Engelenburg, Christophe Garban, Romain Panis, Franco Severo
Abstract
We study the probability that the origin is connected to the boundary of the box of size (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 with wired boundary conditions, we prove that this probability decays as in dimensions , and as when .
- For the infinite volume FK-Ising measure, we prove that this probability decays as in dimensions , and as when .
- For the sourceless double random current measure, we prove that this probability decays as in dimensions , and as when .
Additionally, for the infinite volume FK-Ising measure, we show that the one-arm probability is in dimension , and at least in dimension . This establishes that the FK-Ising model has upper-critical dimension equal to , in contrast to the Ising model, where it is known to be less or equal to , thus solving a conjecture of Chayes, Coniglio, Machta, and Shtengel.