Critical properties of island perimeters in the flooding transition of stochastic and rotational sandpile models

Ahmed, Jahir Abbas; Santra, S. B.

doi:10.1016/j.physa.2012.06.026

Cited by 3 publications

(5 citation statements)

References 57 publications

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“…Note that the anomalous scaling resulted here due to the system size dependence of the correlation function C L,p (r) [35]. Finally, the critical exponent of toppling surfaces and that of the avalanche size distribution are found to be related as D s (p) = 2 + χ(p) [34] at each value of p. The results obtained in two different methods are then consistent.…”

Section: Properties Of Toppling Surfacessupporting

confidence: 52%

“…Third, comparing the values of χ(p) and H(p) for all values p, it is observed that χ(p) ≈ H(p) + 1/2, which suggests that ζ ≈ 1. In that case, the values of χ(p) and H(p) obtained in the RRSM for different values of p do not satisfy the usual Family-Vicsek scaling (no difference in the roughness exponent and the Hurst exponent) [33], rather they satisfy an anomalous scaling given by χ(p) = ζ/2 + H(p) with ζ = 1 for any value of p. Such a scaling relation is already verified for the RSM and the SSM [14,34] independently. Note that the anomalous scaling resulted here due to the system size dependence of the correlation function C L,p (r) [35].…”

Section: Properties Of Toppling Surfacesmentioning

confidence: 90%

“…The dashed lines in Fig. 8 [14,34] independently. Note that the anomalous scaling resulted here due to the system size dependence of the correlation function C L,p (r) [35].…”

Section: Properties Of Toppling Surfacesmentioning

confidence: 98%

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Crossover from rotational to stochastic sandpile universality in the random rotational sandpile model

2014

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In the rotational sandpile model, either the clockwise or the anti-clockwise toppling rule is assigned to all the lattice sites. It has all the features of a stochastic sandpile model but belongs to a different universality class than the Manna class. A crossover from rotational to Manna universality class is studied by constructing a random rotational sandpile model and assigning randomly clockwise and anti-clockwise rotational toppling rules to the lattice sites. The steady state and the respective critical behaviour of the present model are found to have a strong and continuous dependence on the fraction of the lattice sites having the anti-clockwise (or clockwise) rotational toppling rule. As the anti-clockwise and clockwise toppling rules exist in equal proportions, it is found that the model reproduces critical behaviour of the Manna model. It is then further evidence of the existence of the Manna class, in contradiction with some recent observations of the non-existence of the Manna class.

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Section: Properties Of Toppling Surfacessupporting

confidence: 52%

Section: Properties Of Toppling Surfacesmentioning

confidence: 90%

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Crossover from rotational to stochastic sandpile universality in the random rotational sandpile model

2014

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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: N Finitementioning

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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“…In order to characterize various geometrical properties of avalanche one needs to visualize the avalanche in a suitable parameter space. The values of the toppling number S φ [i] of an avalanche at different nodes of SWN define a surface called toppling surface [30] which not only serves as an important quantity to visualize an avalanche but also presents important scaling behaviour of several geometrical properties of the avalanche [31,32]. For an intermediate value of φ (SWN regime), the toppling surfaces of DSSM for both 1d and 2d are presented in Fig.…”

Section: Toppling Surface: Fragmentation Compactness and Fluctuationmentioning

confidence: 99%

Stochastic sandpile model on small-world networks: Scaling and crossover

Bhaumik

Santra

2018

Physica A: Statistical Mechanics and its Applications

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A dissipative stochastic sandpile model is constructed on one and two dimensional smallworld networks with different shortcut densities φ where φ = 0 and 1 represent a regular lattice and a random network respectively. In the small-world regime (2 −12 ≤ φ ≤ 0.1), the critical behaviour of the model is explored studying different geometrical properties of the avalanches as a function of avalanche size s. For both the dimensions, three regions of s, separated by two crossover sizes s 1 and s 2 (s 1 < s 2 ), are identified analyzing the scaling behaviour of average height and area of the toppling surface associated with an avalanche. It is found that avalanches of size s < s 1 are compact and follow Manna scaling on the regular lattice whereas the avalanches with size s > s 1 are sparse as they are on network and follow mean-field scaling. Coexistence of different scaling forms in the small-world regime leads to violation of usual finite-size scaling, in contrary to the fact that the model follows the same on the regular lattice as well as on the random network independently. Simultaneous appearance of multiple scaling forms are characterized by developing a coexistence scaling theory. As SWN evolves from regular lattice to random network, a crossover from diffusive to super-diffusive nature of sand transport is observed and scaling forms of such crossover is developed and verified.

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Critical properties of island perimeters in the flooding transition of stochastic and rotational sandpile models

Cited by 3 publications

References 57 publications

Crossover from rotational to stochastic sandpile universality in the random rotational sandpile model

Crossover from rotational to stochastic sandpile universality in the random rotational sandpile model

Recent advances in percolation theory and its applications

Stochastic sandpile model on small-world networks: Scaling and crossover

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