arXiv subjects

Vincent Tassion

Publications and source records attributed to Vincent Tassion.

At least 37 records · Page 2Linked to original sources

arXiv2018

Subcritical phase of $d$-dimensional Poisson-Boolean percolation and its vacant set

We prove that the Poisson-Boolean percolation on $\mathbb{R}^d$ undergoes a sharp phase transition in any dimension under the assumption that the radius distribution has a $5d-3$ finite moment (in particular we do not assume that the distribution is bounded). More precisely, we prove that: -In the whole subcritical regime, the expected size of the cluster of the origin is finite, and furthermore we obtain bounds for the origin to be connected to distance $n$: when the radius distribution has a finite exponential moment, the probability decays exponentially fast in $n$, and when the radius distribution has heavy tails, the probability is equivalent to the probability that the origin is covered by a ball going to distance $n$. - In the supercritical regime, it is proved that the probability of the origin being connected to infinity satisfies a mean-field lower bound. The same proof carries on to conclude that the vacant set of Poisson-Boolean percolation on $\mathbb{R}^d$ undergoes a sharp phase transition. This paper belongs to a series of papers using the theory of randomized algorithms to prove sharpness of phase transitions.

math.PR↗

arXiv2018

Emergent Planarity in two-dimensional Ising Models with finite-range Interactions

The known Pfaffian structure of the boundary spin correlations, and more generally order-disorder correlation functions, is given a new explanation through simple topological considerations within the model's random current representation. This perspective is then employed in the proof that the Pfaffian structure of boundary correlations emerges asymptotically at criticality in Ising models on $\mathbb Z^2$ with finite-range interactions. The analysis is enabled by new results on the stochastic geometry of the corresponding random currents. The proven statement establishes an aspect of universality, seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases.

math-ph↗

arXiv2018

A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model

We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime $β<β_c$, and the mean-field lower bound $\mathbb{P}_β[0\longleftrightarrow\infty]\ge (β-β_c)/β$ for $β>β_c$. For finite-range models, we also prove that for any $β<β_c$, the probability of an open path from the origin to distance $n$ decays exponentially fast in $n$. For the Ising model, we prove finiteness of the susceptibility for $β<β_c$, and the mean-field lower bound $\langle σ_0\rangle_β^+\ge \sqrt{(β^2-β_c^2)/β^2}$ for $β>β_c$. For finite-range models, we also prove that the two-point correlations functions decay exponentially fast in the distance for $β<β_c$.

math.PR↗

arXiv2017

Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$

We prove that the $q$-state Potts model and the random-cluster model with cluster weight $q>4$ undergo a discontinuous phase transition on the square lattice. More precisely, we show - Existence of multiple infinite-volume measures for the critical Potts and random-cluster models, - Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and - Exponential decay of correlations for the measure with free boundary conditions for both the critical Potts and random-cluster models. The proof is based on a rigorous computation of the Perron-Frobenius eigenvalues of the diagonal blocks of the transfer matrix of the six-vertex model, whose ratios are then related to the correlation length of the random-cluster model. As a byproduct, we rigorously compute the correlation lengths of the critical random-cluster and Potts models, and show that they behave as $\exp(π^2/\sqrt{q-4})$ as $q$ tends to 4.

math.PR↗

arXiv2017

A note on Schramm's locality conjecture for random-cluster models

In this note, we discuss a generalization of Schramm's locality conjecture to the case of random-cluster models. We give some partial (modest) answers, and present several related open questions. Our main result is to show that the critical inverse temperature of the Potts model on $\mathbb Z^r\times(\mathbb Z/2n\mathbb Z)^{d-r}$ (with $r\ge3$) converges to the critical inverse temperature of the model on $\mathbb Z^d$ as $n$ tends to infinity. Our proof relies on the infrared bound and, contrary to the corresponding statement for Bernoulli percolation, does not involve renormalization arguments.

math.PR↗

arXiv2017

Existence of an unbounded vacant set for subcritical continuum percolation

We consider the Poisson Boolean percolation model in $\mathbb{R}^2$, where the radii of each ball is independently chosen according to some probability measure with finite second moment. For this model, we show that the two thresholds, for the existence of an unbounded occupied and an unbounded vacant component, coincide. This complements a recent study of the sharpness of the phase transition in Poisson Boolean percolation by the same authors. As a corollary it follows that for Poisson Boolean percolation in $\mathbb{R}^d$, for any $d\ge2$, finite moment of order $d$ is both necessary and sufficient for the existence of a nontrivial phase transition for the vacant set.

math.PR↗

arXiv2017

Exponential decay of connection probabilities for subcritical Voronoi percolation in $\mathbb{R}^d$

We prove that for Voronoi percolation on $\mathbb{R}^d$, there exists $p_c\in[0,1]$ such that - for $p 0$ such that $\mathbb{P}_p[0\text{ connected to distance }n]\leq \exp(-c_p n)$, - there exists $c>0$ such that for $p>p_c$, $\mathbb{P}_p[0\text{ connected to }\infty]\geq c(p-p_c)$. For dimension 2, this result offers a new way of showing that $p_c(2)=1/2$. This paper belongs to a series of papers using the theory of algorithms to prove sharpness of the phase transition.

math.PR↗

arXiv2016

The Bethe ansatz for the six-vertex and XXZ models: an exposition

In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $ψ$ and energy $Λ$, which satisfy $V ψ= Λψ$, where $V$ is the the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights $a= b=1$ and $c > 0$. We also show that the same vector $ψ$ satisfies $H ψ= E ψ$, where $H$ is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value $E$ computed explicitly. Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors that amounts to proving that the random cluster model on $\mathbb{Z}^2$ with cluster weight $q >4$ exhibits a first-order phase transition.

math.PR↗

arXiv2016

The box-crossing property for critical two-dimensional oriented percolation

We consider critical oriented Bernoulli percolation on the square lattice $\mathbb{Z}^2$. We prove a Russo-Seymour-Welsh type result which allows us to derive several new results concerning the critical behavior: - We establish that the probability that the origin is connected to distance $n$ decays polynomially fast in $n$. - We prove that the critical cluster of the origin conditioned to survive to distance $n$ has a typical width $w_n$ satisfying $εn^{2/5} < w_n < n^{1-ε}$ for some $ε> 0$. The sub-linear polynomial fluctuations contrast with the supercritical regime where $w_n$ is known to behave linearly in $n$. It is also different from the critical picture obtained for non-oriented Bernoulli percolation, in which the scaling limit is non-degenerate in both directions. All our results extend to the graphical representation of the one-dimensional contact process.

math.PR↗

arXiv2016

Sharpness of the phase transition for continuum percolation in R^2

We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood when the radii are uniformly bounded from above. In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter $λ_c$. Moreover, - For $λ< λ_c$, the vacant set has a unique unbounded connected component and we give precise bounds on the one-arm probability for the occupied set, depending on the radius distribution. - At criticality, we establish the box-crossing property, implying that no unbounded component can be found, neither in the occupied nor the vacant sets. We provide a polynomial decay for the probability of the one-arm events, under sharp conditions on the distribution of the radius. - For $λ> λ_c$, the occupied set has a unique unbounded component and we prove that the one-arm probability for the vacant decays exponentially fast. The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation.

math.PR↗

arXiv2016

A new computation of the critical point for the planar random-cluster model with $q\ge1$

We present a new computation of the critical value of the random-cluster model with cluster weight $q\ge 1$ on $\mathbb{Z}^2$. This provides an alternative approach to the result of Beffara and Duminil-Copin. We believe that this approach has several advantages. First, most of the proof can easily be extended to other planar graphs with sufficient symmetries. Furthermore, it invokes RSW-type arguments which are not based on self-duality. And finally, it contains a new way of applying sharp threshold results which avoid the use of symmetric events and periodic boundary conditions.

math.PR↗

arXiv2015

Critical Percolation and the Minimal Spanning Tree in Slabs

The minimal spanning forest on $\mathbb{Z}^{d}$ is known to consist of a single tree for $d \leq 2$ and is conjectured to consist of infinitely many trees for large $d$. In this paper, we prove that there is a single tree for quasi-planar graphs such as $\mathbb{Z}^{2}\times {\{0,\ldots,k\}}^{d-2}$. Our method relies on generalizations of the "Gluing Lemma" of arXiv:1401.7130. A related result is that critical Bernoulli percolation on a slab satisfies the box-crossing property. Its proof is based on a new Russo-Seymour-Welsh type theorem for quasi-planar graphs. Thus, at criticality, the probability of an open path from $0$ of diameter $n$ decays polynomially in $n$. This strengthens the result of arXiv:1401.7130, where the absence of an infinite cluster at criticality was first established.

math.PR↗

arXiv2015

Crossing probabilities for Voronoi percolation

We prove that the standard Russo-Seymour-Welsh theory is valid for Voronoi percolation. This implies that at criticality the crossing probabilities for rectangles are bounded by constants depending only on their aspect ratio. This result has many consequences, such as the polynomial decay of the one-arm event at criticality.

math.PR↗

arXiv2015

Homogenization via sprinkling

We show that a superposition of an $\varepsilon$-Bernoulli bond percolation and any everywhere percolating subgraph of $\mathbb Z^d$, $d\ge 2$, results in a connected subgraph, which after a renormalization dominates supercritical Bernoulli percolation. This result, which confirms a conjecture of the first author together with Häggström and Schramm (2000), is mainly motivated by obtaining finite volume characterizations of uniqueness for general percolation processes.

math.PR↗

arXiv2015

Continuity of the phase transition for planar random-cluster and Potts models with $1\le q\le4$

This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic $q$-state Potts model on $\mathbb Z^2$ is continuous for $q\in\{2,3,4\}$, in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions. The proof uses the random-cluster model with cluster-weight $q\ge1$ (note that $q$ is not necessarily an integer) and is based on two ingredients: 1. The fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights $1\le q\le 4$, which is derived studying parafermionic observables on a discrete Riemann surface. 2. A new result proving the equivalence of several properties of critical random-cluster models: - the absence of infinite-cluster for wired boundary conditions, - the uniqueness of infinite-volume measures, - the sub-exponential decay of the two-point function for free boundary conditions, - a Russo-Seymour-Welsh type result on crossing probabilities in rectangles with arbitrary boundary conditions. The result leads to a number of consequences concerning the scaling limit of the random-cluster model with $1\le q \le 4$. It shows that the family of interfaces (for instance for Dobrushin boundary conditions) are tight when taking the scaling limit and that any sub-sequential limit can be parametrized by a Loewner chain. We also study the effect of boundary conditions on these sub-sequential limits. Let us mention that the result should be instrumental in the study of critical exponents as well.

math.PR↗

arXiv2015

A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb Z^d$

We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that - for $p p_c$, the probability that the origin belongs to an infinite cluster satisfies the mean-field lower bound $θ(p)\ge\tfrac{p-p_c}{p(1-p_c)}$. This note presents the argument of \cite{DumTas15}, which is valid for long-range Bernoulli percolation (and for the Ising model) on arbitrary transitive graphs in the simpler framework of nearest-neighbour Bernoulli percolation on $\mathbb Z^d$.

math.PR↗

arXiv2015

Quenched Voronoi percolation

We prove that the probability of crossing a large square in quenched Voronoi percolation converges to 1/2 at criticality, confirming a conjecture of Benjamini, Kalai and Schramm from 1999. The main new tools are a quenched version of the box-crossing property for Voronoi percolation at criticality, and an Efron-Stein type bound on the variance of the probability of the crossing event in terms of the sum of the squares of the influences. As a corollary of the proof, we moreover obtain that the quenched crossing event at criticality is almost surely noise sensitive.

math.PR↗

arXiv2014

Absence of infinite cluster for critical Bernoulli percolation on slabs

We prove that for Bernoulli percolation on a graph $\mathbb{Z}^2\times\{0,\dots,k\}$ ($k\ge 0$), there is no infinite cluster at criticality, almost surely. The proof extends to finite range Bernoulli percolation models on $\mathbb{Z}^2$ which are invariant under $π/2$-rotation and reflection.

math.PR↗