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