Random geometry and quantum gravity

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

12 févr. 2025, 16:15

1h

Amphi Choquet-Bruhat (batiment Perrin) (Institut Henri Poincaré)

Orateur

Léonie Papon (Durham University, UK)

Description

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 $f(\delta )\to \mathrm{\infty }$, then the interface degenerates to a boundary arc.