Is Critical 2D Percolation Universal?

Beffara, Vincent

doi:10.1007/978-3-7643-8786-0_3

Cited by 10 publications

(35 citation statements)

References 15 publications

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“…Smirnov has then outlined a proof for the conformal invariance of the full percolation configuration. For more technical details of proof we refer to the original paper by Smirnov [80] or [270,271,272].…”

Section: Scaling Limit and Conformal Invariance Of Percolationmentioning

confidence: 99%

Recent advances in percolation theory and its applications

2015

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Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component. Despite its very simple rules, percolation theory has successfully been applied to describe a large variety of natural, technological and social systems. Percolation models serve as important universality classes in critical phenomena characterized by a set of critical exponents which correspond to a rich fractal and scaling structure of their geometric features. We will first outline the basic features of the ordinary model.Over the years a variety of percolation models has been introduced some of which with completely different scaling and universal properties from the original model with either continuous or discontinuous transitions depending on the control parameter, dimensionality and the type of the underlying rules and networks. We will try to take a glimpse at a number of selective variations including Achlioptas process, half-restricted process and spanning cluster-avoiding process as examples of the so-called explosive percolation. We will also introduce non-self-averaging percolation and discuss correlated percolation and bootstrap percolation with special emphasis on their recent progress. Directed percolation process will be also discussed as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state.In the past decade, after the invention of stochastic Löwner evolution (SLE) by Oded Schramm, two-dimensional (2D) percolation has become a central problem in probability theory leading to the two recent Fields medals. After a short review on SLE, we will provide an overview on existence of the scaling limit and conformal invariance of the critical percolation. We will also establish a connection with the magnetic models based on the percolation properties of the Fortuin-Kasteleyn and geometric spin clusters. As an application we will discuss how percolation theory leads to the reduction of the 3D criticality in a 3D Ising model to a 2D critical behavior.Another recent application is to apply percolation theory to study the properties of natural and artificial landscapes. We will review the statistical properties of the coastlines and watersheds and their relations with percolation. Their fractal structure and compatibility with the theory of SLE will also be discussed. The present mean sea level on Earth will be shown to coincide with the critical threshold in a percolation description of the global topography.

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Section: Scaling Limit and Conformal Invariance Of Percolationmentioning

confidence: 99%

Recent advances in percolation theory and its applications

2015

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“…Remark 2. They are various ways to construct a canonical embedding of a planar map, see [9]. However, we work here with Riemann's uniformization because it is well-suited to define and use the SLE 6 exploration (see below).…”

Section: The Riemann Surface Constructionmentioning

confidence: 99%

A Glimpse of the Conformal Structure of Random Planar Maps

Curien

2014

Commun. Math. Phys.

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We present a way to study the conformal structure of random planar maps. The main idea is to explore the map along an SLE (Schramm-Loewner evolution) process of parameter κ = 6 and to combine the locality property of the SLE 6 together with the spatial Markov property of the underlying lattice in order to get a non-trivial geometric information. We follow this path in the case of the conformal structure of random triangulations with a boundary.Under a reasonable assumption called ( * ) that we have unfortunately not been able to verify, we prove that the limit of uniformized random planar triangulations has a fractal boundary measure of Hausdorff dimension 1 3 almost surely. This agrees with the physics KPZ predictions and represents a first step towards a rigorous understanding of the links between random planar maps and the Gaussian free field (GFF)."uniformization" − −−−−−−−−−−−− → Figure 1: A random triangulation embedded (not isometrically) in R 3 and an approximation of its uniformization on the two-dimensional sphere.

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“…Radial SLE (6) and chordal SLE(6) are in fact very closely related, and this property is specific to the case κ = 6: radial SLE(6) and chordal SLE(6) are basically the same, up to the time at which the curve disconnect the target points of the two processes from one another. This is similar to the locality property of SLE (6) (Proposition 3.4).…”

Section: Relation Between Radial and Chordal Sle(6)mentioning

confidence: 91%

“…Our goal is to prove that γ is an SLE (6). The first thing to check is that γ can indeed be almost surely constructed via a Loewner chain.…”

Section: Loewner Chainsmentioning

confidence: 99%