Near-critical Ising interfaces in the thermal direction: Absolute continuity with respect to SLE$_3$

Abstract

The scaling limits of interfaces in the critical two-dimensional Ising model are described by the Schramm--Loewner evolution $\mathrm{SLE}_3$. While, in the scaling limit, the law of the whole near-critical FK-Ising configuration is known to be singular with respect to the critical law in the thermal direction, and the same is expected for the spin configuration, we prove in this paper that the law of a single interface is not. More precisely, we consider the Ising model with mesh $δ$ in the Dobrushin square at $β=β_c-mδ$, where $m\ne0$ is fixed, and we prove that the Radon-Nikodym derivative of the law of the interface with respect to its critical law has moments of every positive and negative order, bounded uniformly in $δ$, so that the law of every subsequential scaling limit of the interface is mutually absolutely continuous with respect to that of $\mathrm{SLE}_3$. Our analysis explains the mechanism behind this result: the logarithmically growing specific heat (that is, the variance of the Hamiltonian, normalized by the number of edges) cancels between the full domain and the domain slit along the interface.