Interface scaling limit for the critical planar Ising model perturbed by a magnetic field

Abstract

We prove that the interface separating $+1$ and $-1$ spins in the critical planar Ising model with Dobrushin boundary conditions perturbed by an external magnetic field has a scaling limit. This result holds when the Ising model is defined on a bounded and simply connected subgraph of $δ\mathbb{Z}^2$, with $δ>0$. We show that if the scaling of the external field is of order $δ^{15/8}$, then, as $δ\to 0$, the interface converges in law to a random curve whose law is conformally covariant and absolutely continuous with respect to SLE$_3$. This limiting law is a massive version of SLE$_3$ in the sense of Makarov and Smirnov and we give an explicit expression for its Radon-Nikodym derivative with respect to SLE$_3$. We also prove that if the scaling of the external field is of order $δ^{15/8}g_1(δ)$ with $g_1(δ)\to 0$, then the interface converges in law to SLE$_3$. In contrast, we show that if the scaling of the external field is of order $δ^{15/8}g_2(δ)$ with $g_2(δ) \to \infty$, then the interface degenerates to a boundary arc.