### Antal A. Járai: Research interests

General areas of interests

Mathematical models of statistical physics, critical phenomena, self-organized criticality, mean-field behaviour

Specific areas of interest
PhD projects available

A PhD project is available related to the Abelian sandpile model, that is described below.

Abelian sandpile model. Let G be a finite connected graph with a distinguished vertex s, called the "sink". We denote the set of vertices that are different from s by V and by E the set of edges. A sandpile on G is an assignment of indistinguishable particles to the vertices in V, specified by a map η : V → {0,1,2,...}. If the vertex x has at least as many particles as its degree, that is, η(x) ≥ deg(x), then x is allowed to topple, which means that x sends one particle to each of its neighbours. Note that we do not keep track of particles reaching the sink. The sandpile η is called stable, if no vertex can topple, that is η(x) < deg(x) for all x ∈ V. It is easy to see (using that G is connected) that any sandpile can be stablized in finitely many steps, by carrying out possible topplings in some order. It can also be shown that the final stable configuration does not depend on what sequence of topplings was used [Dh90]. This is the reason the model is called "Abelian". We define a Markov chain on the set of stable sandpiles as follows. At time step n, we add a particle at a vertex in V chosen uniformly at random, and stabilize, if necessary. It can be shown that there is a unique recurrent class, and that the stationary distribution ν(G) is uniformly distributed on the set of recurrent states [Dh90].

The set of recurrent states, denoted R(G), forms an Abelian group, called the sandpile group of G [Dh90]. The group operation is as follows: given two recurrent sandpiles η, ξ ∈ R(G), we form the sandpile η + ξ (adding componentwise), and then stabilize, to arrive at a new recurrent sandpile. The sandpile group is isomorphic to a factor of the free Abelian group of rank |V|, by the sublattice spanned by the integer row span of a matrix Δ. This matrix Δ is the graph Laplacian on V: Δ(x,y) = deg(x), if x = y; Δ(x,y) = -1, if x is a neighbour of y; and Δ(x,y) = 0 otherwise.

A fascinating feature of the Abelian sandpile model is the behaviour of its avalanches. When we add a new particle in the sandpile Markov chain described above, the sequence of topplings created in stablizing is called an avalanche. Computer simulations (that are not difficult to write) show that the sizes of avalanches range from very small (affecting none or only a few vertices) to very large (affecting almost all vertices). On the other hand, it is a difficult problem in general to prove theorems making such statements quantitative. See [LP10] for some fascinating questions about sandpiles.

Project: Zero dissipation limit. Let V be a finite subset of the d-dimensional integer lattice. Modify the sandpile model to allow continuous height variables η(x) ∈ [0,∞). Let γ > 0 be a real parameter. Call a sandpile stable, if η(x) ∈ [0, 2d + γ). If η(x) ≥ 2d + γ, x is allowed to topple, which means that it sends unit height to each of its neighbours and sends height γ to the sink. In other words, the height of x is reduced by 2d + γ, the height of each neighbour is increased by 1, and amount γ is lost (dissipated). As γ → 0, the discrete sandpile model is recovered [JRS11]. The rate of convergence was estimated in [AJ11] in dimensions d = 2 and 3. The goal of the project is to handle other dimensions. Useful tools for this problem will be a bijection between recurrent sandpiles on G and spanning trees of G [MD92], as well as Wilson's algorithm described under Project 2. Input from the preprint [JW11] will also be useful.

References

[AJ04] Athreya, S.R. and Járai, A.A.: Infinite volume limit for the stationary distribution of Abelian sandpile models. Comm. Math. Phys. 249 197-213 (2004).

[Dh90] Dhar, D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64 1613-1616 (1990).

[J11] Járai, A.A.: Rate of convergence estimates for the zero-dissipation limit in Abelian sandpiles. Preprint (2011). arXiv:1101.1437v2

[JRS11] Járai, A.A., Redig, F. and Saada, E.: Zero dissipation limit in the abelian avalanche model. Preprint (2011). arXiv:0906.3128v3

[JW11] Járai, A.A. and Werning, N.: Minimal configurations and sandpile measures. Preprint (2011). arXiv:1110.4523v1

[La99] Lawler, G.F.: Loop-erased random walk. In Perplexing problems in probability, 197-217, Progr. Probab., 44, Birkhäuser Boston, Boston, MA (1999).

[LP10] Levine, L. and Propp, J.: What is ... a sandpile? Notices Amer. Math. Soc. 57, no. 8, 976-979 (2010).

[LP] Lyons, R. and Peres, Y.: Probability on trees and networks. Book in preparation.

[MD92] Majumdar, S.N. and Dhar, D.: Equivalence between the Abelian sandpile model and the q → 0 limit of the Potts model. Phys. A 185, 129-145 (1992).

[Wi96] Wilson, D.B.: Generating random spanning trees more quickly than the cover time. Proceedings of the twenty-eighth annual ACM Symposium on Theory of computing., ACM, New York, 269-303 (1996).

Research Interests | Publications and Preprints
Home | Department of Mathematical Sciences