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
|