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Register a new userSandpile model on Scale Free Networks with preferential sand distribution: a new universality class
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A two state sandpile model with preferential sand distribution is developed and studied numerically on scale free networks with power-law degree ($k$) distribution, {em i.e.}: $P_ksim k^{-alpha}$. In this model, upon toppling of a critical node sand grains are given one to each of the neighbouring nodes with highest and lowest degrees instead of two randomly selected neighbouring nodes as in a stochastic sandpile model. The critical behaviour of the model is determined by characterizing various avalanche properties at the steady state varying the network structure from scale free to random, tuning $alpha$ from $2$ to $5$. The model exhibits mean field scaling on the random networks, $alpha>4$. However, in the scale free regime, $2<alpha<4$, the scaling behaviour of the model not only deviates from the mean-field scaling but also the exponents describing the scaling behaviour are found to decrease continuously as $alpha$ decreases. In this regime, the critical exponents of the present model are found to be different from those of the two state stochastic sandpile model on similar networks. The preferential sand distribution thus has non-trivial effects on the sandpile dynamics which leads the model to a new universality class.
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We introduce a network growth model in which the preferential attachment probability includes the fitness vertex and the Euclidean distance between nodes. We grow a planar network around its barycenter. Each new site is fixed in space by obeying a power law distribution.
The two-dimensional ($2d$) fully frustrated Planar Rotator model on a square lattice has been the subject of a long controversy due to the simultaneous $Z_2$ and $O(2)$ symmetry existing in the model. The $O(2)$ symmetry being responsible for the Berezinskii - Kosterlitz - Thouless transition ($BKT$) while the $Z_2$ drives an Ising-like transition. There are arguments supporting two possible scenarios, one advocating that the loss of $Ising$ and $BKT$ order take place at the same temperature $T_{t}$ and the other that the $Z_2$ transition occurs at a higher temperature than the $BKT$ one. In the first case an immediate consequence is that this model is in a new universality class. Most of the studies take hand of some order parameter like the stiffness, Binders cumulant or magnetization to obtain the transition temperature. Considering that the transition temperatures are obtained, in general, as an average over the estimates taken about several of those quantities, it is difficult to decide if they are describing the same or slightly separate transitions. In this paper we describe an iterative method based on the knowledge of the complex zeros of the energy probability distribution to study the critical behavior of the system. The method is general with advantages over most conventional techniques since it does not need to identify any order parameter emph{a priori}. The critical temperature and exponents can be obtained with good precision. We apply the method to study the Fully Frustrated Planar Rotator ($PR$) and the Anisotropic Heisenberg ($XY$) models in two dimensions. We show that both models are in a new universality class with $T_{PR}=0.45286(32)$ and $T_{XY}=0.36916(16)$ and the transition exponent $ u=0.824(30)$ ($frac{1}{ u}=1.22(4)$).
A new classification of sandpile models into universality classes is presented. On the basis of extensive numerical simulations, in which we measure an extended set of exponents, the Manna two state model [S. S. Manna, J. Phys. A 24, L363 (1991)] is found to belong to a universality class of random neighbor models which is distinct from the universality class of the original model of Bak, Tang and Wiesenfeld [P. Bak, C. Tang and K. Wiensenfeld, Phys. Rev. Lett. 59, 381 (1987)]. Directed models are found to belong to a universality class which includes the directed model introduced and solved by Dhar
This paper has been withdrawn by the authors due to a possible inconsistency in the program code.
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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