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Résumé |
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. |
