van Engelenburg, D. G. P., Garban, C., Panis, R., & Severo, F. (2025). One-arm exponents of the high-dimensional Ising model. arXiv. https://doi.org/10.48550/arXiv.2510.23423
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.
