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

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 SLE $_3$ . This limiting law is a massive version of SLE $_3$ in the sense of Makarov and Smirnov and I will give an explicit expression for its Radon-Nikodym derivative with respect to SLE $_3$ . I will 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, I will 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.

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

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