arXiv subjects

Stéphane Benoist

Publications and source records attributed to Stéphane Benoist.

6 recordsLinked to original sources

arXiv2019

Near-critical spanning forests and renormalization

We study random two-dimensional spanning forests in the plane that can be viewed both in the discrete case and in their appropriately taken scaling limits as a uniformly chosen spanning tree with some Poissonian deletion of edges or points. We show how to relate these scaling limits to a stationary distribution of a natural coalescent-type Markov process on a state-space of abstract graphs with real-valued edge-weights. This Markov process can be interpreted as a renormalization flow. This provides a model for which one can rigorously implement the formalism proposed by the third author in order to relate the law of the scaling limit of a critical model to a stationary distribution of such a renormalization/Markov process: When starting from any two-dimensional lattice with constant edge-weights, the Markov process does indeed converge in law to this stationary distribution that corresponds to a scaling limit of UST with Poissonian deletions. The results of this paper heavily build on the convergence in distribution of branches of the UST to SLE$_2$ (a result by Lawler, Schramm and Werner) as well as on the convergence of the suitably renormalized length of the loop-erased random walk to the "natural parametrization" of the SLE$_2$ (a recent result by Lawler and Viklund).

math.PR↗

arXiv2018

The scaling limit of critical Ising interfaces is CLE(3)

In this paper, we consider the set of interfaces between + and - spins arising for the critical planar Ising model on a domain with + boundary conditions, and show that it converges towards CLE(3). Our proof relies on the study of the coupling between the Ising model and its random cluster (FK) representation, and of the interactions between FK and Ising interfaces. The main idea is to construct an exploration process starting from the boundary of the domain, to discover the Ising loops and to establish its convergence to a conformally invariant limit. The challenge is that Ising loops do not touch the boundary; we use the fact that FK loops touch the boundary (and hence can be explored from the boundary) and that Ising loops in turn touch FK loops, to construct a recursive exploration process that visits all the macroscopic loops. A key ingredient in the proof is the convergence of Ising free arcs to the Free Arc Ensemble (FAE), established in [BDH16]. Qualitative estimates about the Ising interfaces then allow one to identify the scaling limit of Ising loops as a conformally invariant collection of simple, disjoint SLE(3)-like loops and thus by the Markovian characterization of [ShWe12] as a CLE(3). A technical point of independent interest contained in this paper is an investigation of double points of interfaces in the scaling limit of critical FK-Ising. It relies on the technology of [KeSm12].

math.PR↗

arXiv2017

Natural parametrization of SLE: the Gaussian free field point of view

We provide another construction of the natural parametrization of SLE$_κ$ for $κ< 4$. We construct it as the expectation of the quantum time, which is a random measure carried by SLE in an ambient Gaussian free field. This quantum time was built as the push forward on the SLE curve of the Liouville boundary measure, which is a natural field-dependent measure supported on the boundary of the domain. We moreover show that the quantum time can be reconstructed as a chaos on any measure on the trace of SLE with the right Markovian covariance property. This provides another proof that natural parametrization is characterized by its Markovian covariance property.

math.PR↗

arXiv2016

Classifying conformally invariant loop measures

We formulate a classification conjecture for conformally invariant families of measures on simple loops that builds on a conjecture of Kontsevich and Suhov. The main example in this class of objects was constructed by Werner as boundaries of Brownian loops. We present partial results towards the algebraic step of this classification. Solving this conjecture would provide another argument explaining why planar statistical mechanics models with conformally invariant scaling limits naturally occur in a one-parameter family, together with the dynamical characterization of SLE via Schramm's central limit argument, and with the conformal field theory point of view and its central charge parameter.

math-ph↗

arXiv2016

Conformal invariance of crossing probabilities for the Ising model with free boundary conditions

We prove that crossing probabilities for the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, a phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin. We do so by establishing the convergence of certain exploration processes towards SLE$(3,\frac{-3}2,\frac{-3}2)$. We also construct an exploration tree for free boundary conditions, analogous to the one introduced by Sheffield.

math.PR↗

arXiv2015

An SLE$_2$ loop measure

There is an essentially unique way to associate to any Riemann surface a measure on its simple loops, such that the collection of measures satisfy a strong conformal invariance property. Wendelin Werner constructed these random simple loops as boundaries of Brownian loops, and so they correspond in the zoo of statistical mechanics models to central charge $0$, or Schramm-Loewner Evolution (SLE) parameter $κ=8/3$. The goal of this paper is to construct a family of measures on simple loops on Riemann surfaces that satisfies a conformal covariance property, and that would correspond to SLE parameter $κ=2$ (central charge $-2$). On planar annuli, this loop measure was already built by Adrien Kassel and Rick Kenyon. We will give an alternative construction of this loop measure on planar annuli, investigate its conformal covariance, and finally extend this measure to general Riemann surfaces. This gives an example of a Malliavin-Kontsevich-Suhov loop measure in non-zero central charge.

math.PR↗