The recurrent states of the Abelian sandpile model (ASM) are those states that appear infinitely often. For this reason they occupy a central position in ASM research. We present several new results for classifying recurrent states of the Abelian sandpile model on graphs that may be decomposed in a variety of ways. These results allow us to classify, for certain families of graphs, recurrent st...

We report our findings of a 1 f power spectrum for the total amount of sand in directed and undirected Bak-Tang-Wiesenfeld models confined to narrow stripes and driven locally. The underlying mechanism for the 1 f noise in these systems is an exponentially long configuration memory giving rise to a very broad distribution of time scales. Both models are solved analytically with the help of an o...

We numerically study avalanches in the two-dimensional Abelian sandpile model in terms of a sequence of waves of toppling events. Priezzhev et al. @Phys. Rev. Lett. 76, 2093 ~1996!# have recently proposed exact results for the critical exponents in this model based on the existence of a proposed scaling relation for the difference in sizes of subsequent waves, Ds5sk2sk11 , where the size of the...

A brief historical perspective is first given concerning financial crashes, from the 17th till the 20th century. In modern times, it seems that log periodic oscillations are found before crashes in several financial indices. The same is found in sand pile avalanches on Sierpinski gaskets. A discussion pertains to the after shock period with illustrations from the DAX index. The factual financia...

The Abelian Sandpile Model is a discrete di usion process de ned on graphs (Dhar [14], Dhar et al. [15]) which serves as the standard model of self-organized criticality. The transience class of a sandpile is de ned as the maximum number of particles that can be added without making the system recurrent ([4]). We develop the theory of discrete di usions in contrast to continuous harmonic functi...

We prove that the one-dimensional sandpile prediction problem is in AC. The previously best known upper bound on the AC-scale was AC. We also prove that it is not in AC1− for any constant

We point out a new mechanism which can lead to mean field type behaviour in nonequilibrium critical phenomena. We demonstrate this mechanism on a two-dimensional model which can be understood as a stochas-tic and non-conservative version of the abelian sandpile model of Bak et al. [1]. This model has a second order phase transition whose critical behaviour seems at least partly described by the...

We present limiting shape results for a non-abelian variant of the abelian sandpile growth model (ASGM), some of which have no analog in the ASGM. One of our limiting shapes is an octagon. In our model, mass spreads from the origin by the toppling rule in Zhang’s sandpile model. Previously, several limiting shape results have been obtained for the ASGM using abelianness and monotonicity as main...

We consider a variational model that describes the growth of a sandpile on a bounded table under the action of a vertical source. The possible equilibria of such a model solve a boundary value problem for a system of nonlinear partial differential equations that we analyse when the source term is merely integrable. In addition, we study the asymptotic behavior of the dynamical problem showing t...

We discuss a general framework for cutting constructions and reinterpret in this setting the work on non-Abelian symplectic cuts by Weitsman. We then introduce two analogous non-Abelian modification constructions for hyperkähler manifolds: one modifies the topology significantly, the other gives metric deformations. We highlight ways in which the geometry of moment maps for non-Abelian hyperkäh...