One-arm exponents of the high-dimensional Ising model

TL;DR

This paper analyzes the decay of one-arm probabilities in high-dimensional Ising-related models, revealing different universality classes and establishing the upper-critical dimension for FK-Ising.

Contribution

It proves the decay rates of one-arm probabilities across various models and dimensions, confirming the upper-critical dimension of FK-Ising as 6, resolving a longstanding conjecture.

Findings

Decay rates of one-arm probabilities in different models and dimensions

Identification of the upper-critical dimension for FK-Ising as 6

Contrast with the Ising model's critical dimension

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 $1/n^{1+o(1)}$ 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.