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Exact solution of stochastic directed sandpile model

We introduce and analytically solve a directed sandpile model with stochastic toppling rules. The model clearly belongs to a different universality class from its counterpart with deterministic toppling rules, previously solved by Dhar and Ramaswamy. The critical exponents are D_||=7/4, \tau=10/7 in two dimensions and D_||=3/2, \tau=4/3 in one dimension. The upper critical dimension of the model is three, at which the exponents apart from logarithmic corrections reach their mean-field values D_||=2, \tau=3/2.

Numerical and analytical studies of sandpile models of Self-Organized Criticality [1] continue to be a subject of considerable research activity. In particular, a lot of effort has been recently invested in establishing the set of universality classes in these systems [2]. The consensus seems to be that the universality class of a d-dimensional sandpile model is determined by answers to the following list of questions: (i) Is it a critical slope or a critical height model? In other words, does a site topple when its local slope or its height exceeds a certain threshold value. Critical height models (e.g. Bak-Tang-Wiesenfeld (BTW) model [1]) were studied more extensively in the past and are in general better understood.
(ii) Is sand redistributed isotropically in a toppling event? According to this property sandpiles could be classified as isotropic or directed (anisotropic). The common knowledge is that it is a relevant parameter, i.e. an arbitrary small anisotropy of toppling rules usually drives the model to the directed universality class.
(iii) Finally, it is important if the sand is distributed deterministically or randomly in each individual toppling. In a model with deterministic toppling rules the configuration of the bulk of the sandpile remains unchanged if every single site on the lattice topples exactly once. This additional symmetry is usually important for the universality class of the model: e.g. the deterministic one-dimensional isotropic critical height sandpile (1D BTW) has only trivially distributed avalanches of fractal dimension 2, while its cousins with additional randomness in toppling rules, such as the Zaitsev model [3], Oslo model [4], Linear Interface Model [5], etc., seem to belong to the same universality class where avalanches have a non-integer fractal dimension D ≃ 2.23 and a probability distribution with a clean power law exponent τ ≃ 1.27.
Despite much numerical and analytical effort on the original BTW sandpile model (which is a deterministic/isotropic/critical-height model in the above classification), its critical exponents in two dimen-sions still remain controversial [2]. The situation is somewhat better for directed models. Soon after the original BTW sandpile model [1], Dhar and Ramaswamy introduced and exactly solved in all dimensions its directed cousin -the Dhar-Ramaswamy (DR) model [6].
Both BTW and DR models have deterministic toppling rules. As far as stochastic models are concerned there is preciously little analytical results. Apart from an exact solution of a model, equivalent to the 1D stochastic directed sandpile [7], stochastic sandpiles were studied only numerically. In this paper we present an analytical study of the stochastic directed sandpile model in all dimensions. Stochastic directed sandpiles were brought to the attention of the community in two recent papers [8], reporting numerical studies of several variants of such models in two dimensions. During the preparation of this manuscript, there appeared a closely related preprint by Paczuski and Bassler [9] in which an analytical study of the directed stochastic sandpile model was presented and similar results were obtained. In particular, using different analytical arguments they have arrived at the same set of exponents.
The microscopic rules of the stochastic directed sandpile model that we selected to study are closely related to those of the DR model [6]. These rules are modified in the spirit of a stochastic isotropic sandpile model known as the Manna model [10]. It is easier to define our rules in two dimensions, while generalization to higher dimensions is straightforward. A stable configuration of our model is specified by the integer height of the sandpile z(x 1 , x 2 ) ≤ 1 at each point of a 2D square lattice. The lattice has open boundary conditions along the diagonal coordinate, x || = x 1 + x 2 , and periodic boundary conditions in the transversal direction x ⊥ = x 1 − x 2 . The sand is added randomly at the line with x || = 0 and falls off the edge at x || = L || . The difference between our model and the DR model lies in toppling rules. In both cases, once the height at any given site exceeds one, this site becomes unstable and loses two grains of sand to its nearest neighbors in the direction of increasing x || . How-ever, while in the DR model each of these two neighbors gets exactly one grain of sand, in our stochastic variant the decision where to move any particular grain is done independently for each grain. In other words, with probability 1/4 both grains end up on the left neighbor, with probability 1/4 they go to the right neighbor, and only with probability 1/2 will each neighbor get one grain as in the DR model. Obviously, on average each neighbor gets one grain, yet the additional stochastic element in these rules drives the model away from the universality class of the DR model [8]. It is easy to see that unlike the deterministic rules of the DR model, the new stochastic rules allow for multiple topplings of some sites within one avalanche. Indeed, let us consider an example where the first active site of the avalanche toppled one grain of sand to each of its two neighbors. Let us further assume that these neighbors both had z = 1 so that they both toppled and by chance distributed all four of the resulting grains of sand to the same site in the next layer. This site has received four grains of sand and is guaranteed to topple twice. The numerical simulations [8] confirm the existence of multiple topplings in other variants of a stochastic directed sandpile model.
In order to get an analytical handle on the properties of our model we employ the same trick that was used by one of us to solve the 1D directed stochastic sandpile model [7]. Due to the Abelian nature of the model, we can change the order in which topplings are performed without changing the outcome. It is convenient to do topplings layer by layer. This means that we topple any given site as many times as necessary to make it stable before moving on to the next unstable site, and we topple all unstable sites in one layer (a given x || ) before toppling any site in the next layer. Let us concentrate on a site with coordinates x || and x ⊥ immediately after we have finished with topplings in the (x || − 1)-th layer. Assume that two of its neighbors with coordinates x || − 1 and x ⊥ ± 1 have toppled, respectively, n 1 = n(x || − 1, x ⊥ − 1) and n 2 = n(x || − 1, x ⊥ + 1) times. The average number of grains of sand that our selected site would receive from the previous layer is (n 1 + n 2 ). In the DR model there are no fluctuations around this average. Also, due to the absence of multiple topplings in this deterministic directed model, n 1 and n 2 can be only 0 or 1. Therefore, in the DR model a site can receive either n 1 + n 2 = 2 grains, in which case it is guaranteed to topple exactly once, or n 1 + n 2 = 1, in which case it can topple with probability 1/2 (i.e. it topples if it had z = 1 before the transfer and remains stable if it had z = 0). From this one can show [6] that in the DR model the set of sites which topple at each layer form an interval with no holes inside. The size of this interval as a function of the layer number x || performs an ordinary random walk.
In the stochastic model the relation between the number of topplings in two subsequent layers is more complicated. Let us concentrate on the behavior of the total number of topplings N (x || ) = x ⊥ n(x || , x ⊥ ) in a given layer x || . This number fixes the number of grains of sand transferred from the layer x || to the next layer as 2N (x || ). It is easy to see that a site which has received an even number 2k of grains of sand from a previous layer will always topple exactly k times and, therefore, will transfer the same 2k grains of sand to the layer directly below it. That means that as far as N (x || ) is concerned, such sites behave in a completely passive manner, i.e. they do not lead to a decrease or an increase of the total number of topplings N (x || ) from layer to layer. On the other hand, any site which received an odd number 2k + 1 of grains of sand from a previous layer has equal chance to topple k times (if it had z = 0 before the transfer) or k+1 times (if it had z = 1). In the former case this site would decrease the grain flow 2N (x || ) by one, while in the latter -increase by 1. Let us call any site which has received an odd number of grains of sand from the previous layer an active site. The equation, which is the central result of this work, relating the change in the total number of topplings from layer to layer to the number of active sites N a (x || ) in a given layer is N ( ξ a , where all ξ a are −1 or +1 with equal probability and independent of each other. These random numbers correspond to whether each of the N a (x || ) active sites had the height z = 0 or z = 1 before the avalanche started. It is straightforward to demonstrate that, as in the DR model, in the steady state of the directed stochastic model all possible stable configurations of z are equally represented, and, therefore, there are no correlations between the heights at different sites, and each height is equally likely to have z = 0 or z = 1. It is more convenient to rewrite the above equation in a continuous notation, which works as long as N a (x || ) >> 1 : Here η(t) is a standard Gaussian variable with zero mean and the standard deviation equal to unity. This equation describes an unbiased random walk N (x || ) vs x || with a variable step, given by 1 2 N a (x || ). A random walk (an avalanche) starts with N (0) = 1 and ends at x || when N (x || ) ≤ 0 for the first time. Let us assume that both N (x || ) and N a (x || ) in a surviving avalanche scale with x || with the exponents α and α a , respectively. Alternatively one can say that they are related to each other by N a ∼ N αa/α . Plugging this relation into Eq. (1), after easy algebra we get the exponent relation In addition to this exponent relation, the mapping of the avalanche process onto a random walk immediately gives us the power law τ || in the probability distribution of avalanche sizes. Indeed, an avalanche ends when the random walk described by Eq. (1) first enters the N (x || ) ≤ 0 semi-axis, and it is a standard result of the theory of stochastic processes that the distribution of first returns of a generalized random walk has an exponent τ || = 1 + α. This agrees with the well-known exponent relation τ || = D || valid for a general directed sandpile model. Indeed, the fractal dimension D || , defined by || , is obviously related to α through D || = 1 + α.
The Eq. (1) applies equally well to the DR and the stochastic directed sandpile models. The difference between these two models lies only in the scaling of the number of active sites with x || . As was explained above, in the 2D DR model the only two active sites lie at the edge of an interval of toppled sites. Indeed, only these sites get 1 grain of sand, while the rest get either 0 or 2. Therefore, in the 2D DR model N a (x || ) = 2 is just a constant, α a = 0, and the Eq. (1) describes an ordinary random walk, in which N ( || . The introduction of a stochastic element in particle redistribution dramatically changes the number of active sites at any given layer of the avalanche. Indeed, when grains are distributed independently, any site which has at least one toppled neighbor in the previous layer is equally likely to receive an even or odd number of grains of sand, and therefore, it has a probability 1/2 of becoming active. Thus, in the stochastic model the exponent α a defines how the number of distinct sites that topple at least once, scales with the layer number x || . The difference between exponents α and α a comes solely from the existence of multiple topplings. These two exponents have to obey the inequality α ≥ α a , and their difference α − α a determines how the average number of topplings n top (x || ) at a given site in x || -th layer scales with x || : We proceed with an argument that in the 2D directed stochastic model α a = 1/2, and, therefore, by the virtue of Eq. (2) α = 3/4. It is a straightforward task to determine the average number of topplings n(x || , x ⊥ ) at a given site x || , x ⊥ , where the average is performed over the whole ensemble of avalanches, so that avalanches that die out before reaching this site, contribute 0 to the average. As was noted in [6], due to the conservation of sand and the stationarity of the sandpile n(x || , x ⊥ ) has to satisfy the diffusion equation with a source: This average balance equation is also exact for our stochastic model, where it proves that, like in the DR model, sites that topple at least once are spread over the interval of length ∆x ⊥ ∼ x 1/2 || in the x || layer. In the DR model these sites form a dense interval with no holes, and, therefore, their number is known to scale exactly as x 1/2 || . The situation is somewhat less obvious in the stochastic directed model, where the set of toppled sites can have holes. However, one can argue that these holes would mostly be concentrated near the boundaries of the avalanche in any given layer, while the majority of toppled sites in the center would form a compact interval. Indeed, as will be confirmed later, the 2D stochastic directed model is characterized by multiple topplings, where a site at a layer x || would typically topple n top (x || ) ∼ x 1/4 || times within one avalanche. Since any of the 2n top grains is equally likely to go to each of the two nearest neighbors in the next layer, the situation where one of these neighbors would receive less than two grains is exponentially unlikely for large n top . But once a site has received two or more grains of sand it is guaranteed to topple at least once. Therefore, both nearest neighbors down the slope from the site which toppled many times would most likely topple at least once. In other words, the creation of a new hole (a region free of topplings) is exponentially suppressed near sites which themselves toppled many times. This means that for a sufficiently large x || the majority of sites (especially those close to the center of the avalanche region) will belong to a hole-free region. Since the size of the interval covered by an avalanche scales as x  [8] and with our own simulations of the model. In Fig. 1 we present the results of our simulations for the effective exponents α = d log N/d log x || and α a = d log N a /d log x || as a function of x || . The numerical exponent α nicely agrees with the analytical results. The exponent α a is less clean due to the presence of holes near the boundary of the avalanche region. The exponent seems first to overshoot to a value of almost 0.6 but then goes down so that in the end of the range of our simulations, x || ∼ 30000, it is consistent with our theoretical prediction α a = 1/2.
Unlike its deterministic cousin, the stochastic directed sandpile model exhibits a non-trivial scaling even in one dimension. The 1D deterministic directed sandpile model is trivial in the sense that there is just one SOC configuration and the addition of a grain of sand always results in an avalanche of L || topplings in which this grain is transported and discarded at the open boundary of the system. A 1D stochastic directed sandpile model can have several variants of simple microscopic toppling rules. In one variant, which is essentially identical to the model studied by one of us in [7], once a height at a given site exceeds one, either one or two grains are transferred to the nearest neighbor down the slope. It is easy to see (for details see [7]) that this model is equivalent to a 1D random walk so that N a = const, while the typical number of topplings N scales as a function of x || = x as N (x) ∼ x 1/2 . The distribution of avalanche spatial length in this model has an exponent τ || = 3/2, while that of avalanche volumeτ = 4/3. In another variant of the 1D stochastic toppling rules an unstable site always loses two grains of sand and each of these grains with equal probability goes to the nearest or next-nearest neighbors down the slope.
As in the DR model the upper critical dimension for the stochastic directed sandpile model is d u = 3. In this dimension the expected number of topplings at each site in a layer x || grows only logarithmically with x || . Therefore, α = α a apart from the logarithmic corrections. From Eq. (1) in this case we get α = (1 + α)/2, which has the solution α = 1, τ || = 2, τ = 3/2. This is a standard set of mean-field exponents for any branching (avalanche) process in high enough dimension. In Fig. 2 we plot the numerical effective exponents in the 3D stochastic directed model. They agree well with the mean field values.
In conclusion, we have found an analytic solution of the stochastic directed sandpile model in any dimension. The main difference of this model from its deterministic counterpart -the Dhar-Ramaswamy model -lies in the fractal dimension of the set of active sites, i.e sites that can add or remove one grain from the overall flow of sand between two subsequent layers. Whereas in the 2D DR model in any layer there are only two sites at the edges of the interval of toppled sites which are active, in the 2D stochastic directed sandpile model each of approximately x 1/2 || toppled sites in this interval has a 1/2 chance of being active in the above sense. This leads to an increase in the fractal dimension of an avalanche from D || = 3/2 to D || = 7/4 due to the appearance of multiple topplings. The difference between critical properties of stochastic and deterministic directed models disappears in high dimensions d ≥ 3, where multiple topplings in a stochastic directed sandpile become prohibitively unlikely and all exponents acquire their mean-field values. The scale dependent effective exponents α = d log N/d log x || and αa = d log Na/d log x || as a function of x || for the 2D model, and for two longitudinal system sizes L || = 5000 and L || = 40000.