arXiv subjects

Vincent Beffara

Publications and source records attributed to Vincent Beffara.

At least 19 recordsLinked to original sources

arXiv2025

A new bound for the critical point of the FK model for $q<1$

We consider the random cluster model with parameter $q<1$, for which the FKG inequalities are not valid. On the square lattice, stochastic comparison with Bernoulli percolation implies that the model is subcritical (respectively supercritical) when $p \leq q/(1+q)$ (resp. $p \geq 1/2$); in this paper, we extend these two regions, by improving the classical stochastic comparisons. Assuming the existence of the critical point, this reduces its possible range. The proof relies on a modification of the usual Glauber dynamics of the model, which enables stochastic bounds of FK measures between two inhomegenous percolations. We also prove uniqueness of the infinite-volume measure in our extended ranges. Most of our results are valid in any dimension $d \geq 2$ and beyond hypercubic lattices.

math.PR↗

arXiv2020

Local Geometry of the rough-smooth interface in the two-periodic Aztec diamond

Random tilings of the two-periodic Aztec diamond contain three macroscopic regions: frozen, where the tilings are deterministic; rough, where the correlations between dominoes decay polynomially; smooth, where the correlations between dominoes decay exponentially. In a previous paper, the authors found that a certain averaging of height function differences at the rough-smooth interface converged to the extended Airy kernel point process. In this paper, we augment the local geometrical picture at this interface by introducing well-defined lattice paths which are closely related to the level lines of the height function. We show, after suitable centering and rescaling, that a point process from these paths converges to the extended Airy kernel point process provided that the natural parameter associated to the two-periodic Aztec diamond is small enough.

math.PR↗

arXiv2019

Trees of self-avoiding walks

We consider the biased random walk on a tree constructed from the set of finite self-avoiding walks on a lattice, and use it to construct probability measures on infinite self-avoiding walks. The limit measure (if it exists) obtained when the bias converges to its critical value is conjectured to coincide with the weak limit of the uniform SAW. Along the way, we obtain a criterion for the continuity of the escape probability of a biased random walk on a tree as a function of the bias, and show that the collection of escape probability functions for spherically symmetric trees of bounded degree is stable under uniform convergence.

math.PR↗

arXiv2019

Scaling limits for random triangulations on the torus

We study the scaling limit of essentially simple triangulations on the torus. We consider, for every $n\geq 1$, a uniformly random triangulation $G_n$ over the set of (appropriately rooted) essentially simple triangulations on the torus with $n$ vertices. We view $G_n$ as a metric space by endowing its set of vertices with the graph distance denoted by $d_{G_n}$ and show that the random metric space $(V(G_n),n^{-1/4}d_{G_n})$ converges in distribution in the Gromov-Hausdorff sense when $n$ goes to infinity, at least along subsequences, toward a random metric space. One of the crucial steps in the argument is to construct a simple labeling on the map and show its convergence to an explicit scaling limit. We moreover show that this labeling approximates the distance to the root up to a uniform correction of order $o(n^{1/4})$.

cs.DM↗

arXiv2018

On the Uniqueness of Global Multiple SLEs

This article focuses on the characterization of global multiple Schramm-Loewner evolutions (SLE). The chordal SLE describes the scaling limit of a single interface in various critical lattice models with Dobrushin boundary conditions, and similarly, global multiple SLEs describe scaling limits of collections of interfaces in critical lattice models with alternating boundary conditions. In this article, we give a minimal amount of characterizing properties for the global multiple SLEs: we prove that there exists a unique probability measure on collections of pairwise disjoint continuous simple curves with a certain conditional law property. As a consequence, we obtain the convergence of multiple interfaces in the critical Ising, FK-Ising, and percolation models.

math.PR↗

arXiv2017

Airy point process at the liquid-gas boundary

Domino tilings of the two-periodic Aztec diamond feature all of the three possible types of phases of random tiling models. These phases are determined by the decay of correlations between dominoes and are generally known as solid, liquid and gas. The liquid-solid boundary is easy to define microscopically and is known in many models to be described by the Airy process in the limit of a large random tiling. The liquid-gas boundary has no obvious microscopic description. Using the height function we define a random measure in the two-periodic Aztec diamond designed to detect the long range correlations visible at the liquid-gas boundary. We prove that this random measure converges to the extended Airy point process. This indicates that, in a sense, the liquid-gas boundary should also be described by the Airy process.

math.PR↗

arXiv2017

Percolation without FKG

We prove a Russo-Seymour-Welsh theorem for the antiferromagnetic Ising model with small parameter on a regular enough periodic planar triangulation. More generally we prove that the box-crossing property extends to a neighborhood of the product measure among certain families of discrete percolation models that do not necessarily satisfy the Fortuin-Kasteleyn-Ginibre condition of positive association.

math.PR↗

arXiv2016

Percolation of random nodal lines

We prove a Russo-Seymour-Welsch percolation theorem for nodal domains and nodal lines associated to a natural infinite dimensional space of real analytic functions on the real plane. More precisely, let $U$ be a smooth connected bounded open set in $\mathbb R^2$ and $γ, γ'$ two disjoint arcs of positive length in the boundary of $U$. We prove that there exists a positive constant $c$, such that for any positive scale $s$, with probability at least $c$ there exists a connected component of $\{x\in \bar U, \, f(sx) \textgreater{} 0\} $ intersecting both $γ$ and $γ'$, where $f$ is a random analytic function in the Wiener space associated to the real Bargmann-Fock space. For $s$ large enough, the same conclusion holds for the zero set $\{x\in \bar U, \, f(sx) = 0\} $. As an important intermediate result, we prove that sign percolation for a general stationary Gaussian field can be made equivalent to a correlated percolation model on a lattice.

math.PR↗

arXiv2015

On the critical parameters of the $q\ge4$ random-cluster model on isoradial graphs

The critical surface for random-cluster model with cluster-weight $q\ge 4$ on isoradial graphs is identified using parafermionic observables. Correlations are also shown to decay exponentially fast in the subcritical regime. While this result is restricted to random-cluster models with $q\ge 4$, it extends the recent theorem of the two first authors to a large class of planar graphs. In particular, the anisotropic random-cluster model on the square lattice is shown to be critical if $\frac{p_vp_h}{(1-p_v)(1-p_h)}=q$, where $p_v$ and $p_h$ denote the horizontal and vertical edge-weights respectively. We also mention consequences for Potts models.

math.PR↗

arXiv2015

Dessins d'enfants for analysts

We present an algorithmic way of exactly computing Belyi functions for hypermaps and triangulations in genus 0 or 1, and the associated dessins, based on a numerical iterative approach initialized from a circle packing combined with subsequent lattice reduction. The main advantage compared to previous methods is that it is applicable to much larger graphs; we use very little algebraic geometry, and aim for this paper to be as self-contained as possible.

math.CV↗

arXiv2014

Critical percolation on mesoscopic triangulations

We extend Smirnov's proof of the existence and conformal invariance of the scaling limit of critical site-percolation on the triangular lattice to particular sequences of periodic graphs with more arbitrary large-scale structure, obtained by piecing together triangular regions of the triangular lattice. While not formally speaking a scaling limit statement (as the graphs are not rescaled versions of each other), the result is a weak form of universality for critical percolation.

math.PR↗

arXiv2013

Bridges and random truncations of random matrices

We continue to study the squared Frobenius norm of a submatrix of a $n \times n$ random unitary matrix. When the choice of the submatrix is deterministic and its size is $[ns] \times [nt]$, we proved in a previous paper that, after centering and without any rescaling, the two-parameter process converges in distribution to a bivariate Brownian bridge. Here, we consider Bernoulli independent choices of rows and columns with respective parameters $s$ and $t$. We prove by subordination that after centering and rescaling by $n^{-1/2}$, the process converges to another Gaussian process.

math.PR↗

arXiv2013

The self-dual point of the two-dimensional random-cluster model is critical for $q\geq 1$

We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter $q\geq1$ on the square lattice is equal to the self-dual point $p_{sd}(q) = \sqrt q /(1+\sqrt q)$. This gives a proof that the critical temperature of the $q$-state Potts model is equal to $\log (1+\sqrt q)$ for all $q\geq 2$. We further prove that the transition is sharp, meaning that there is exponential decay of correlations in the sub-critical phase. The techniques of this paper are rigorous and valid for all $q\geq 1$, in contrast to earlier methods valid only for certain given $q$. The proof extends to the triangular and the hexagonal lattices as well.

math.PR↗

arXiv2013

On the critical value function in the divide and color model

The divide and color model on a graph $G$ arises by first deleting each edge of $G$ with probability $1-p$ independently of each other, then coloring the resulting connected components (\emph{i.e.}, every vertex in the component) black or white with respective probabilities $r$ and $1-r$, independently for different components. Viewing it as a (dependent) site percolation model, one can define the critical point $r_c^G(p)$. In this paper, we mainly study the continuity properties of the function $r_c^G$, which is an instance of the question of locality for percolation. Our main result is the fact that in the case $G=\mathbb Z^2$, $r_c^G$ is continuous on the interval $[0,1/2)$; we also prove continuity at $p=0$ for the more general class of graphs with bounded degree. We then investigate the sharpness of the bounded degree condition and the monotonicity of $r_c^G(p)$ as a function of $p$.

math.PR↗

arXiv2013

Confidence intervals for the critical value in the divide and color model

We obtain confidence intervals for the location of the percolation phase transition in Häggström's divide and color model on the square lattice $\mathbb{Z}^2$ and the hexagonal lattice $\mathbb{H}$. The resulting probabilistic bounds are much tighter than the best deterministic bounds up to date; they give a clear picture of the behavior of the DaC models on $\mathbb{Z}^2$ and $\mathbb{H}$ and enable a comparison with the triangular lattice $\mathbb{T}$. In particular, our numerical results suggest similarities between DaC model on these three lattices that are in line with universality considerations, but with a remarkable difference: while the critical value function $r_c(p)$ is known to be constant in the parameter $p$ for $p<p_c$ on $\mathbb{T}$ and appears to be linear on $\mathbb{Z}^2$, it is almost certainly non-linear on $\mathbb{H}$.

math.PR↗

arXiv2013

Planar percolation with a glimpse of Schramm-Loewner Evolution

In recent years, important progress has been made in the field of two-dimensional statistical physics. One of the most striking achievements is the proof of the Cardy-Smirnov formula. This theorem, together with the introduction of Schramm-Loewner Evolution and techniques developed over the years in percolation, allow precise descriptions of the critical and near-critical regimes of the model. This survey aims to describe the different steps leading to the proof that the infinite-cluster density $θ(p)$ for site percolation on the triangular lattice behaves like $(p-p_c)^{5/36+o(1)}$ as $p\searrow p_c=1/2$.

math.PR↗

arXiv2012

On monochromatic arm exponents for 2D critical percolation

We investigate the so-called monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well understood) polychromatic exponents. More specifically, our main result is that the monochromatic j-arm exponent is strictly between the polychromatic j-arm and (j+1)-arm exponents.

math.PR↗

arXiv2009

On a randomized PNG model with a columnar defect

We study a variant of poly-nuclear growth where the level boundaries perform continuous-time, discrete-space random walks, and study how its asymptotic behavior is affected by the presence of a columnar defect on the line. We prove that there is a non-trivial phase transition in the strength of the perturbation, above which the law of large numbers for the height function is modified.

math.PR↗