Workshop on

**"Sandpile models and
related fields"**

**September 10-13, 2007**

**EURANDOM, Eindhoven, The Netherlands**

**ABSTRACTS**

**Rob van den Berg (CWI, Amsterdam)
**(Most of this talk is based on joint work with Yuval Peres,
Vladas Sidoravicius and Eulalia Vares)

**Random growth with paralyzing obstacles**

We study models of random spatial growth processes where initially there are sources of growth (indicated by the colour green) and sources of a growth-stopping (paralyzing) substance (red). The green sources expand and may merge with others (there is no `inter-green' competition). The red substance remains passive as long as it is isolated. However, when a green cluster comes in touch with the red substance, it is immediately invaded by the latter, stops growing and starts to act as red substance itself. In our main model space is represented by a graph, of which initially each vertex is randomly green, red or white (vacant), and the growth of the green clusters is similar to that in first-passage percolation.

One of the main issues we investigated is the size of a green cluster just before it is paralyzed. This research also led to a new result for invasion percolation.

**Robert Cori (Université Bordeaux)**

**About the structure of the abelian
group defined by the sandpile model on graphs **

**Deepak Dhar (Tata Institute Bombay)**

**Sandpile Models: where do we stand today?**

**Anne Fey (EURANDOM, VU Amsterdam) **

**Sandpile percolation**

As a variant on the usual dynamical sandpile model, we study so-called sandpile toppling processes in a parametrized sandpile model. In this model, we perform sandpile topplings ordered in time, on translation invariant initial configurations in . The parameter is the density of sand; for small , the toppling process has a stable limit. In this limit configuration, we then study percolation of sites that toppled at least once in the toppling process. We find a subcritical regime, that is, there exist such that the toppled cluster size distribution has an exponential tail.

**Alexis Gillet (VU Amsterdam)**

**The Bak-Sneppen Model **

The Bak-Sneppen model is a self-organised critical model for the evolution of interacting species. Despite being one of the more well-known examples of self-organised criticality, the Bak-Sneppen model has largely resisted rigorous analysis. In this presentation, recent progress will be explained with computer simulations used for illustration.

**Antal Jarai (Carleton University)
**Joint work with F. Redig and E. Saada

**The zero-dissipation limit in Abelian sandpiles**

We study Abelian sandpiles on the d-dimensional integer lattice with continuous heights and arbitrary positive dissipation at each site. We show that under a natural condition on the addition rates, the dissipative process converges to the critical sandpile process, as the dissipation goes to zero. Some open problems will also be discussed.

**Lionel Levine (University of California, Berkeley)
**

**The sandpile group of a tree **

I will describe how the sandpile group of a tree relates to the sandpile groups of its principal subtrees, and show how to compute the full decomposition of the sandpile group of a regular tree into cyclic subgroups. I will also give two applications to the rotor-router model. First, I will show that rotor-router aggregation on a regular tree yields a perfect ball when an appropriate number of particles have aggregated. Second, I will characterize the possible "escape sequences" for rotor-router walk on the infinite ternary tree, that is, binary words $a_1 \ldots a_n$ such that the $k$-th particle escapes to infinity without returning to the origin if and only if $a_k=1$.

**
Criel Merino
(Universidad Nacional Autónoma de México)
**

Combinatorial interpretations of the critical configurations in the chip-firing game

The chip-firing game is a solitaire game played in a graph and it is a combinatorial form of the Abelian Sandpile Model of Dhar when the toppling matrix is symmetric and the adding of particles is not random but orderly.

If we classify the critical configurations of the chip-firing game according to its weight and take the ordinary generating function of this new sequence we obtain a polynomial which is a rich combinatorial object. We show that is an evaluation of a well-known invariant in combinatorics called the Tutte polynomial. From this, we obtain several other combinatorial interpretations of the critical configurations.

**Yuval Peres (University of California, Berkeley) **

**Rotor-router aggregation, the abelian sandpile and the divisible
sandpile**

The rotor-router model is a deterministic analogue of random walk proposed by Jim Propp, extending the model of "Eulerian walkers". It can be used to define a growth process analogous to internal DLA. We prove that the asymptotic shape of this model in the Euclidean lattice is a ball, and discuss open problems on the patterns of directions of particle last exits and on the related abelian sandpile model. Indeed, the best way we know to understand the correspondence between recurrent sandpile configurations and spanning trees, is via the action of the sandpile on rotor configurations. I will also discuss the divisible sandpile for which we have the most precise picture. (Joint work with Lionel Levine.)

**Philippe Ruelle (Université
Catholique Louvain)**

**The CFT approach to critical sandpile models**

Conformal Field Theory (CFT) is a powerful technique to study critical lattice models. After a short review/reminder of the 2nd Abelian sandpile model, we will present the basic principles and methods of CFT applied to the description of the sandpile model in its time invariant regime.

**
Ellen Saada (CNRS / Université
de Rouen)
**(Joint work with C. Maes, F. Redig)

**Freezing transitions in
non-Fellerian particle systems.**

Non-Fellerian particle systems are characterized by nonlocal interactions. They exhibit new phenomena unseen in standard interacting particle systems. The sandpile model in infinite volume is a typical example of non-Fellerian particle system, we refer to [1,2,5] for a presentation and properties of the model. Our paper [4] dealt with the sandpile dynamics on , it was the first study of the model in infinite volume, and in particular of its non-Feller property.

We will develop the results of [??0], where we consider freezing transitions in one-dimensional non-Fellerian processes which are built from the abelian sandpile additions to which in one case, spin flips are added, and in another case, so called anti-sandpile subtractions. In the first case and as a function of the sandpile addition rate, there is a sharp transition from a non-trivial invariant measure to the trivial invariant measure of the sandpile process. For the combination sandpile plus anti-sandpile, there is a sharp transition from one frozen state to the other anti-state.

[1] Maes C., *New Trends in
Interacting Particle Systems*, Markov Proc. Rel. Fields , vol. 2, 283–288
(2005).

[2] Maes, C., Redig, F., Saada
E., *Abelian sandpile models in infinite volume*, Sankhya, the Indian
Journal of Statistics, , no. 4, 634–661 (2005).

[3] Maes, C., Redig, F., Saada
E., *Freezing transitions in non-Fellerian particle systems*, J. of Stat.
Phys., available on line (2007).

[4] Maes, C., Redig, F., Saada
E. and Van Moffaert, A., *On the thermodynamic limit for a one-dimensional
sandpile process*, Markov Proc. Rel. Fields, , 1–22 (2000).

[5] Redig, F., *Mathematical
aspects of abelian sandpiles*, Lecture notes for Les Houches Summer school on
mathematical statistical physics, Elsevier (2007).

**Vladas Sidoravicius (CWI Amsterdam)
**Joint work with Ronald Dickman (UFMG)

**Sleepy walkers: facts, conjectures, challenges**

Abstract t.b.a.

**Klaus Schmidt (University of
Vienna, Erwin Schrödinger Institute Vienna)**

**Abelian Sandpiles and the Harmonic Model**

There exists an entropy-preserving equivariant and surjective map from the -dimensional critical sandpile model to a certain closed, shift-invariant subgroup of (the ’harmonic model’). The purpose of this talk is the description of this map and some consequences and open questions related to its construction.

**Rinke J.
Wijngaarden (VU University Amsterdam)**

**Physical sand pile models: rice and
superconductors**

We study sand pile dynamics in two different real-world systems: in the vortex matter in superconductors and in a ~ 1 m high pile of rice with a footprint of 1 x 1 m².

The dynamics of vortices in superconductors can be straightforwardly mapped to pile dynamics by interpreting the vortex density as the height variable. The thus created 'vortexscape' is similar to a mountainscape, with its characteristic roughness and its avalanches. Depending on the temperature, we observe either power law distributed avalanches or large scale thermal run-away avalanches. By reducing the sample thickness, the avalanche physics can be even made non-local (which bears some similarity to the solar flare physics).

In the rice pile dynamics we observe nice power law scaling over more than 3 orders of magnitude if the foot of the pile rests on a flat surface. We find that in addition very large quasi-periodic avalanches are created if the foot of the pile is at a ledge. The latter case corresponds to the situation in many sand pile experiments. We conjecture that the reported difference between sand pile models and sand pile experiments may be due to the same difference. Although apparently the 'ledge' boundary condition applies in both cases, in real experiments the grains at the ledge are much more unstable than elsewhere, contrary to the assumption in the models.

For both experimental systems we determine the characteristic exponents to try and validate the exponent relations by Paczuski Maslow and Bak.

*This page is maintained by **Lucienne
Coolen*