Direct evidence for conformal invariance of avalanche frontiers in sandpile models

Saberi, Abbas Ali; Moghimi-Araghi, S.; Dashti-Naserabadi, H.; Rouhani, S.

doi:10.1103/physreve.79.031121

Cited by 30 publications

(37 citation statements)

References 50 publications

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“…Shcramm-Loewner evolution (SLE), as a technique to process the geometrical properties of 2D critical models, has been helpful as well as CFT to uncover the geometrical aspects of this model. It has been shown that 2D BTW model is SLE with diffusivity parameter κ = 2, i.e., SLE 2 [17]. This result is consistent with the CFT predictions, i.e., the relation between SLE and CFT c = (6−κ)(3κ−8) 2κ [18].…”

Section: Introductionsupporting

confidence: 80%

“…It is easy to see that in any wave, the set of toppled sites forms a connected cluster with no voids (untoppled sites fully surrounded by toppled sites), and no site topples more than once in one wave. Geometrical aspects of this model have been the subject of intense studies recently [17,23,24]. One example is the exterior perimeter of an avalanche or a wave that is numerically shown to be loop-erased random walk (LERW) in two dimensions [17,23].…”

Section: A Wavesmentioning

confidence: 99%

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Statistical investigation of the cross sections of wave clusters in the three-dimensional Bak-Tang-Wiesenfeld model

Dashti-Naserabadi

Najafi

2015

Phys. Rev. E

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We consider the three-dimensional (3D) Bak-Tang-Wiesenfeld model in a cubic lattice. Along with analyzing the 3D problem, the geometrical structure of the two-dimensional (2D) cross section of waves is investigated. By analyzing the statistical observables defined in the cross sections, it is shown that the model in that plane (named as 2D-induced model) is in the critical state and fulfills the finite-size scaling hypothesis. The analysis of the critical loops that are interfaces of the 2D-induced model is of special importance in this paper. Most importantly, we see that their fractal dimension is D(f)=1.387±0.005, which is compatible with the fractal dimension of the external perimeter of geometrical spin clusters of 2D critical Ising model. Some hyperscaling relations between the exponents of the model are proposed and numerically confirmed. We then address the problem of conformal invariance of the mentioned domain walls using Schramm-Lowener evolution (SLE). We found that they are described by SLE with the diffusivity parameter κ=2.8±0.2, nearly consistent with observed fractal dimension.

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Section: Introductionsupporting

confidence: 80%

Section: A Wavesmentioning

confidence: 99%

Statistical investigation of the cross sections of wave clusters in the three-dimensional Bak-Tang-Wiesenfeld model

Dashti-Naserabadi

Najafi

2015

Phys. Rev. E

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“…In 2010, Smirnov was awarded the Fields medal for the proof of conformal invariance of percolation and the planar Ising model in statistical physics. SLE has soon found many applications and turned out to describe the vorticity lines in turbulence [26,86], domain walls of spin glasses [87,88,89], the nodal lines of random wave functions [90,91], the iso-height lines of random grown surfaces [92,93,94,95,96,97], the avalanche lines in sandpile models [98] and the coastlines and watersheds on Earth [99,101,102]. Among which, SLE could provide quite unexpected connections between some features of interacting systems and ordinary uncorrelated percolation [26,90].…”

Section: Introductionmentioning

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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“…Nevertheless, numerically correspondence has been shown for a large number of models as mentioned above. It has been argued that SLE can be applied to models exhibiting nonself-crossing paths on a lattice, showing self-similarity, not only in equilibrium but also out of equilibrium as the example discussed here [8,22,23]. In random discretized landscapes each site is characterized by a real number such as, e.g., the height in an elevation map, the intensity in a pixelated image, or the energy in an energy landscape [14].…”

mentioning

confidence: 99%

Watersheds are Schramm-Loewner Evolution Curves

et al. 2012

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We show that in the continuum limit watersheds dividing drainage basins are Schramm-Loewner Evolution (SLE) curves, being described by one single parameter κ. Several numerical evaluations are applied to ascertain this. All calculations are consistent with SLEκ, with κ = 1.734 ± 0.005, being the only known physical example of an SLE with κ < 2. This lies outside the well-known duality conjecture, bringing up new questions regarding the existence and reversibility of dual models. Furthermore it constitutes a strong indication for conformal invariance in random landscapes and suggests that watersheds likely correspond to a logarithmic Conformal Field Theory (CFT) with central charge c ≈ −7/2.PACS numbers: 89.75. Da, 64.60.al, 91.10.Jf The possibility of statistically describing the properties of random curves with a single parameter fascinates physicists and mathematicians alike. This capability is provided by the theory of Schramm-Loewner Evolution (SLE), where random curves can be generated from a Brownian motion with diffusivity κ [1]. Once κ is identified, several geometrical properties of the curve are known (e.g. fractal dimension, winding angle, and leftpassage probability) [2,3]. Among the examples of such curves, we find self-avoiding walks [4] and the contours of critical clusters in percolation [5], Q-state Potts model [6], and spin glasses [7], as well as in turbulence [8]. Establishing SLE for such systems has provided valuable information on the underlying symmetries and paved the way to some exact results [5,9,10]. In fact, SLE is not a general property of non-self-crossing walks since many curves have been shown not to be SLE as, for example, the interface of solid-on-solid models [11], the domain walls of bimodal spin glasses [12], and the contours of negative-weight percolation [13].Recently, the watershed (WS) of random landscapes [14][15][16], with a fractal dimension d f ≈ 1.22, was shown to be related to a family of curves appearing in different contexts such as, e.g., polymers in strongly disordered media [17], bridge percolation [14], and optimal path cracks [18]. In the present Letter, we show that this universal curve has the properties of SLE, with κ = 1.734 ± 0.005. κ < 2 is a special limit since, up to now, all known examples of SLE found in Nature and statistical physics models have 2 ≤ κ ≤ 8, corresponding to fractal dimensions d f between 1.25 and 2.Scale invariance and, consequently, the appearance of fractal dimensions have always motivated to apply concepts from conformal invariance to shed light on critical systems. Archetypes of self-similarity are the contours of critical clusters in lattice models. Already back in 1923, Loewner proposed an expression for the evolution of an analytic function which conformally maps the region bounded by these curves into a standard domain [19]. Such an evolution, follows the theory, should only depend on a continuous function of a real parameter, known as driving function. Recently, Schramm argued that to guarantee conformal invariance, and...

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Direct evidence for conformal invariance of avalanche frontiers in sandpile models

Cited by 30 publications

References 50 publications

Statistical investigation of the cross sections of wave clusters in the three-dimensional Bak-Tang-Wiesenfeld model

Statistical investigation of the cross sections of wave clusters in the three-dimensional Bak-Tang-Wiesenfeld model

Recent advances in percolation theory and its applications

Watersheds are Schramm-Loewner Evolution Curves

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