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Hugo Duminil-Copin

Publications and source records attributed to Hugo Duminil-Copin.

At least 55 records · Page 3Linked 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

Sharp threshold phenomena in statistical physics

This text describes the content of the Takagi lectures given by the author in Kyoto in 2017. The lectures present some aspects of the theory of sharp thresholds for boolean functions and its application to the study of phase transitions in statistical physics.

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

Sixty years of percolation

Percolation models describe the inside of a porous material. The theory emerged timidly in the middle of the twentieth century before becoming one of the major objects of interest in probability and mathematical physics. The golden age of percolation is probably the eighties, during which most of the major results were obtained for the most classical of these models, named Bernoulli percolation, but it is really the two following decades which put percolation theory at the crossroad of several domains of mathematics. In this short (and very partial) review, we propose to describe briefly some of the recent progress as well as some famous challenges remaining in the field. This review is not intended to probabilists (and a fortiori not to specialists in percolation theory): the target audience is mathematicians of all kinds.

math.PR↗
arXiv2017

Universality for the random-cluster model on isoradial graphs

We show that the canonical random-cluster measure associated to isoradial graphs is critical for all $q \geq 1$. Additionally, we prove that the phase transition of the model is of the same type on all isoradial graphs: continuous for $1 \leq q \leq 4$ and discontinuous for $q > 4$. For $1 \leq q \leq 4$, the arm exponents (assuming their existence) are shown to be the same for all isoradial graphs. In particular, these properties also hold on the triangular and hexagonal lattices. Our results also include the limiting case of quantum random-cluster models in $1+1$ dimensions.

math.PR↗
arXiv2017

Higher order corrections for anisotropic bootstrap percolation

We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with $(1,2)$-neighbourhood and threshold $r = 3$. The first order asymptotics for the critical probability were recently determined by the first and second authors. Here we determine the following sharp second and third order asymptotics: \[ p_c\big( [L]^2,\mathcal{N}_{(1,2)},3 \big) \; = \; \frac{(\log \log L)^2}{12\log L} \, - \, \frac{\log \log L \, \log \log \log L}{ 3\log L} + \frac{\left(\log \frac{9}{2} + 1 \pm o(1) \right)\log \log L}{6\log L}. \] We note that the second and third order terms are so large that the first order asymptotics fail to approximate $p_c$ even for lattices of size well beyond $10^{10^{1000}}$.

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

Random currents expansion of the Ising model

Critical behavior at an order/disorder phase transition has been a central object of interest in statistical physics. In the past century, techniques borrowed from many different fields of mathematics (Algebra, Combinatorics, Probability, Complex Analysis, Spectral Theory, etc) have contributed to a more and more elaborate description of the possible critical behaviors for a large variety of models. The Ising model is maybe one of the most striking success of this cross-fertilization, for this model of ferromagnetism is now very well understood both physically and mathematically. In this article, we review an approach, initiated in \cite{GriHurShe70,Aiz82} and based on the notion of random currents, enabling a deep study of the model.

math-ph↗
arXiv2017

Lectures on the Ising and Potts models on the hypercubic lattice

Phase transitions are a central theme of statistical mechanics, and of probability more generally. Lattice spin models represent a general paradigm for phase transitions in finite dimensions, describing ferromagnets and even some fluids (lattice gases). It has been understood since the 1980s that random geometric representations, such as the random walk and random current representations, are powerful tools to understand spin models. In addition to techniques intrinsic to spin models, such representations provide access to rich ideas from percolation theory. In recent years, for two-dimensional spin models, these ideas have been further combined with ideas from discrete complex analysis. Spectacular results obtained through these connections include the proofs that interfaces of the two-dimensional Ising model have conformally invariant scaling limits given by SLE curves, that the connective constant of the self-avoiding walk on the hexagonal lattice is given by $\sqrt{2+\sqrt 2}$. In higher dimensions, the understanding also progresses with the proof that the phase transition of Potts models is sharp, and that the magnetization of the three-dimensional Ising model vanishes at the critical point.

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↗
arXiv2017

Brochette percolation

We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on $\mathbb{Z}$. Select vertical columns at random independently with a given positive probability. Keep (respectively remove) vertical edges in the selected columns, with probability $p$, (respectively $1-p$). All horizontal edges and vertical edges lying in unselected columns are kept (respectively removed) with probability $q$, (respectively $1-q$). We show that, if $p > p_c(\mathbb{Z}^2)$ (the critical point for homogeneous Bernoulli bond percolation) then $q$ can be taken strictly smaller then $p_c(\mathbb{Z}^2)$ in such a way that the probability that the origin percolates is still positive.

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

Exponential decay of loop lengths in the loop $O(n)$ model with large $n$

The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been conjectured that both the spin and the loop $O(n)$ models exhibit exponential decay of correlations when $n>2$. We verify this for the loop $O(n)$ model with large parameter $n$, showing that long loops are exponentially unlikely to occur, uniformly in the edge weight $x$. Our proof provides further detail on the structure of typical configurations in this regime. Putting appropriate boundary conditions, when $nx^6$ is sufficiently small, the model is in a dilute, disordered phase in which each vertex is unlikely to be surrounded by any loops, whereas when $nx^6$ is sufficiently large, the model is in a dense, ordered phase which is a small perturbation of one of the three ground states.

math-ph↗
arXiv2016

The sharp threshold for the Duarte model

The class of critical bootstrap percolation models in two dimensions was recently introduced by Bollobás, Smith and Uzzell, and the critical threshold for percolation was determined up to a constant factor for all such models by the authors of this paper. Here we develop and refine the techniques introduced in that paper in order to determine a sharp threshold for the Duarte model. This resolves a question of Mountford from 1995, and is the first result of its type for a model with drift.

math.PR↗