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Vincent Tassion

Bio: Vincent Tassion is an academic researcher from ETH Zurich. The author has contributed to research in topics: Percolation & Bernoulli's principle. The author has an hindex of 16, co-authored 58 publications receiving 1031 citations. Previous affiliations of Vincent Tassion include École normale supérieure de Lyon & University of Geneva.

Papers published on a yearly basis

Papers
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Journal ArticleDOI
TL;DR: In this article, the sharpness of the phase transition for Bernoulli percolation and the Ising model was shown for infinite-range models on arbitrary locally finite transitive infinite graphs.
Abstract: 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 $${\beta < \beta_c}$$ , and the mean-field lower bound $${\mathbb{P}_\beta[0\longleftrightarrow \infty ]\ge (\beta-\beta_c)/\beta}$$ for $${\beta > \beta_c}$$ . For finite-range models, we also prove that for any $${\beta < \beta_c}$$ , 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 $${\beta < \beta_c}$$ , and the mean-field lower bound $${\langle \sigma_0\rangle_\beta^+\ge \sqrt{(\beta^2-\beta_c^2)/\beta^2}}$$ for $${\beta > \beta_c}$$ . For finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for $${\beta < \beta_c}$$ .

158 citations

Journal ArticleDOI
TL;DR: In this paper, an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces was shown to be applicable to lattice spin models and their random-cluster representations.
Abstract: We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that 1. For the Potts model on transitive graphs, correlations decay exponentially fast for $\beta<\beta_c$. 2. For the random-cluster model with cluster weight $q\geq1$ on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the mean-field lower bound in the supercritical regime. 3. For the random-cluster models with cluster weight $q\geq1$ on planar quasi-transitive graphs $\mathbb{G}$, $$\frac{p_c(\mathbb{G})p_c(\mathbb{G}^*)}{(1-p_c(\mathbb{G}))(1-p_c(\mathbb{G}^*))}~=~q.$$ As a special case, we obtain the value of the critical point for the square, triangular and hexagonal lattices (this provides a short proof of the result of Beffara and Duminil-Copin [Probability Theory and Related Fields, 153(3-4):511--542, 2012]). These results have many applications for the understanding of the subcritical (respectively disordered) phase of all these models. The techniques developed in this paper have potential to be extended to a wide class of models including the Ashkin-Teller model, continuum percolation models such as Voronoi percolation and Boolean percolation, super-level sets of massive Gaussian Free Field, and random-cluster and Potts model with infinite range interactions.

151 citations

Posted Content
TL;DR: In this article, it was shown that both the critical Potts model and the random-cluster model undergo a discontinuous phase transition on the square lattice, and that the correlation lengths of the two models behave as φ(exp(pi 2/π{q-4})$ as π tends to 4.
Abstract: 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 $\exp(\pi^2/\sqrt{q-4})$ as $q$ tends to 4.

105 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the standard Russo-Seymour-Welsh theory is valid for Voronoi percolation and that at criticality the crossing probabilities for rectangles are bounded by constants depending only on their aspect ratio.
Abstract: 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.

96 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the phase transition of the nearest-neighbor ferromagnetic q-state Potts model is continuous for q ≥ 2,3,4.
Abstract: 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 $${q \ge 1}$$ (note that q is not necessarily an integer) and is based on two ingredients: The result has important consequences toward the study of the scaling limit of the random-cluster model with $${q \in [1,4]}$$ . 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.

87 citations


Cited by
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Journal ArticleDOI
TL;DR: Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944 as mentioned in this paper, and there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them.
Abstract: R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

1,658 citations

01 Aug 1993
TL;DR: One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References as discussed by the authors
Abstract: One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

1,491 citations

Book
20 Jan 2017
TL;DR: In this article, the authors present a state-of-the-art account of probability on networks, including percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks.
Abstract: Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

803 citations

Journal ArticleDOI
TL;DR: In this article, the sharpness of the phase transition for Bernoulli percolation and the Ising model was shown for infinite-range models on arbitrary locally finite transitive infinite graphs.
Abstract: 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 $${\beta < \beta_c}$$ , and the mean-field lower bound $${\mathbb{P}_\beta[0\longleftrightarrow \infty ]\ge (\beta-\beta_c)/\beta}$$ for $${\beta > \beta_c}$$ . For finite-range models, we also prove that for any $${\beta < \beta_c}$$ , 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 $${\beta < \beta_c}$$ , and the mean-field lower bound $${\langle \sigma_0\rangle_\beta^+\ge \sqrt{(\beta^2-\beta_c^2)/\beta^2}}$$ for $${\beta > \beta_c}$$ . For finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for $${\beta < \beta_c}$$ .

158 citations