Near-critical interfaces and massive Schramm-Loewner evolutions

Abstract

In this thesis, we study examples of near-critical interfaces in planar statistical mechanics models. These interfaces are defined on a lattice and we are interested in constructing their scaling limit as the meshsize of the lattice tends to 0. We show that in our examples, the limiting curves are described by massive SLE, as conjectured by Makarov and Smirnov [MS10].
In the first chapter, we look at a model called the massive harmonic explorer. This model was proposed by Makarov and Smirnov as a near-critical, or massive, perturbation of the harmonic explorer [MS10]. They argued that the scaling limit of the massive harmonic explorer in a bounded domain is a massive version of chordal SLE4, called massive SLE4, which is conformally covariant and absolutely continuous with respect to chordal SLE4. We provide a full and rigorous proof of this statement. Moreover, we show that a massive SLE4 curve can be coupled with a massive Gaussian free field as its level line, when the field has appropriate boundary conditions.
In the second chapter, we instigate further the above coupling between a massive GFF and a massive SLE4 curve. We give an alternative proof of this coupling in which the coupling is constructed by reweighting the law of the standard GFF-SLE4 coupling. As a consequence of this construction, we derive an explicit expression for the Radon-Nikodym derivative of the law of massive SLE4 with respect to the law of SLE4. We then prove that by reweighting the law of the coupling GFF-CLE4 in a similar way, one obtains a coupling between a massive GFF and a random countable collection of simple loops, that we call massive CLE4. As in the massless setting, these loops can be seen as the level lines of the massive GFF. Using this coupling, we show that massive CLE4 can in turn be coupled to a massive Brownian loop soup of intensity 1/2 in such way that the loops of massive CLE4 correspond to the outer boundaries of the outermost clusters of loops in the loop soup. This proves a conjecture of Camia [Cam13]. Moreover, this relation between massive CLE4 and the massive Brownian loop soup of intensity 1/2 yields an explicit expression for the Radon-Nikodym derivative of the law of massive CLE4 with respect to that of CLE4. Finally, we note that, as the law of the massive GFF, the laws of massive SLE4 and massive CLE4 are conformally covariant.
In the third chapter, we consider the interface separating +1 and −1 spins in the critical planar Ising model with Dobrushin boundary conditions perturbed by an external magnetic field. We 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 δZ^2, with δ > 0. We show that if the scaling of the external field is of order δ^, then, as δ → 0, the interface converges in law to a random curve whose law is conformally covariant and absolutely continuous with respect to SLE3. This limiting law is a massive version of SLE3 in the sense of Makarov and Smirnov [MS10] and we give an explicit expression for its Radon-Nikodym derivative with respect to SLE3. We also prove that if the scaling of the external field is of order δ^g(δ) with g(δ) → 0, then the interface converges in law to SLE3. In contrast, we show that if the scaling of the external field is of order δ^f(δ) with f(δ) → ∞, then the interface degenerates to a boundary arc.