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

TL;DR

This paper establishes the scaling limit of interfaces in the critical planar Ising model under a magnetic field, showing convergence to a conformally covariant SLE$_3$-related curve when the field scales appropriately.

Contribution

It introduces a massive SLE$_3$ as the scaling limit of Ising interfaces under a magnetic field perturbation, with explicit Radon-Nikodym derivative and different regimes analyzed.

Findings

Convergence to a massive SLE$_3$ when external field scales as δ^{15/8}

Explicit Radon-Nikodym derivative for the limiting law

Degeneration to boundary arc when the external field dominates

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 $\delta \mathbb{Z}^2$, with $\delta >0$. We 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 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 $\delta^{15/8}g_1(\delta)$ with $g_1(\delta)\to 0$, then the interface converges in law to SLE$_3$. In contrast, we show that if the scaling of the external field is of order $\delta^{15/8}g_2(\delta)$ with $g_2(\delta) \to \infty$, then the interface degenerates to a boundary arc.