One-arm exponents of the high-dimensional Ising model

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 $1/n^{1+o(1)}$ when $d=4$. - For the infinite volume FK-Ising measure, we prove that this probability decays as $1/n^2$ in dimensions $d>6$, and as $1/n^{2+o(1)}$ when $d=6$. - For the sourceless double random current measure, we prove that this probability decays as $1/n^{d-2}$ in dimensions $d>4$, and as $1/n^{2+o(1)}$ when $d=4$. Additionally, for the infinite volume FK-Ising measure, we show that the one-arm probability is $1/n^{1+o(1)}$ in dimension $d=4$, and at least $1/n^{3/2}$ 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.

💡 Research Summary

The paper investigates the one‑arm probability at criticality for several percolation‑type representations of the Ising model in high dimensions. The one‑arm event is the event that the origin is connected to the boundary of a box Λₙ =