arXiv subjects

Léonie Papon

Publications and source records attributed to Léonie Papon.

5 recordsLinked to original sources

arXiv2026

Scaling limits of the single-curve interface and outermost loops in the planar random field Ising model

We prove that the interface separating $+1$ and $-1$ spins in the near-critical planar random field Ising model (RFIM) with Dobrushin boundary conditions has a scaling limit, whose law is conformally covariant and almost surely absolutely continuous with respect to SLE$_3$. The limiting curve can be seen as a massive version of SLE$_3$ in the sense of Makarov and Smirnov, but in a random environment. We then show that the outermost spin loops of the near-critical planar RFIM with $+1$ boundary conditions have subsequential limits and that any of these limits is almost surely singular with respect to CLE$_3$. This dichotomy between absolute continuity of the single interface and singularity of the outermost loops reflects the fact that a single interface does not explore enough of the magnetization field of the near-critical RFIM to detect the singularity of this field with respect to the critical Ising magnetization field, whereas the outermost spin loops do.

math.PR↗

arXiv2024

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

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

math.PR↗

arXiv2024

Massive SLE$_4$, massive CLE$_4$ and the massive planar GFF

We construct a coupling between a massive GFF and a random curve in which the curve can be interpreted as the level line of the field and has the law of massive SLE$_4$. This coupling is obtained by reweighting the law of the standard coupling GFF-SLE$_4$ and our result can be seen as a conditional version of the path-integral formulation of the massive GFF. We then show that by reweighting the law of the coupling GFF-CLE$_4$ in a similar way, one obtains a coupling between a massive GFF and a random countable collection of simple loops, that we call massive CLE$_4$. Using this coupling, we relate massive CLE$_4$ to the massive Brownian loop soup with intensity $1/2$, thus proving a conjecture of Camia. As the law of the massive GFF, the laws of massive SLE$_4$ and massive CLE$_4$ are conformally covariant.

math.PR↗

arXiv2023

Massive SLE$_4$ and the scaling limit of the massive harmonic explorer

The massive harmonic explorer is a model of random discrete path on the hexagonal lattice that was proposed by Makarov and Smirnov as a massive perturbation of the harmonic explorer. They argued that the scaling limit of the massive harmonic explorer in a bounded domain is a massive version of chordal SLE$_4$, called massive SLE$_4$, which is conformally covariant and absolutely continuous with respect to chordal SLE$_4$. In this paper, we provide a full and rigorous proof of this statement. Moreover, we show that a massive SLE$_4$ curve can be coupled with a massive Gaussian free field as its level line, when the field has appropriate boundary conditions.

math.PR↗

arXiv2022

Thick points of the planar GFF are totally disconnected for all $γ\ne 0$

We prove that the set of $γ$-thick points of a planar Gaussian free field (GFF) with Dirichlet boundary conditions is a.s. totally disconnected for all $γ\neq 0$. Our proof relies on the coupling between a GFF and the nested CLE$_4$. In particular, we show that the thick points of the GFF are the same as those of the weighted CLE$_4$ nesting field and establish the almost sure total disconnectedness of the complement of a nested CLE$_κ$, $κ\in (8/3,4]$. As a corollary we see that the set of singular points for supercritical LQG metrics is a.s. totally disconnected.

math.PR↗