Self-organized critical phenomena

The concept of self-organized criticality was proposed as an
explanation for the occurrence of fractal structures in diverse
natural phenomena. Roughly speaking the idea behind self-organized
criticality is that a dynamic drives a system towards a stationary
state that is characterized by power law correlations in space and
time. We study two of the most famous models that were introduced
as models exhibiting self-organized criticality. The first of them
is the forest fire model. In a forest fire model each site (vertex)
of a graph is either vacant or occupied by a tree. Vacant sites get
occupied according to independent rate 1 Poisson processes.
Independently, at each sites ignition (by lightning) occurs
according to independent Poisson processes that have rate
Lambda>0. When a site is ignited its whole cluster of occupied
sites becomes vacant instantaneously. It is known that infinite
volume forest fire processes exist for all ignition rates
Lambda>0. The proof of existence is rather abstract, and does
not imply uniqueness. Nor does the construction answer the question
whether infinite volume forest fire processes are measurable with
respect to their driving Poisson processes. Motivated by these
questions, we show the almost sure infinite volume convergence for
forest fire models with respect to their driving Poisson processes.
Our proof is quite general and covers all graphs with bounded
vertex, all positive ignition rates Lambda>0, and a quite large
set of initial configurations. One of the main ingredients of the
proof is an estimate for the decay of the cluster size distribution
in a forest fire model. For Gamma>0, we study the probability
that the cluster at site x and time t>=Gamma is larger than m,
conditioned on the configuration of some further clusters at time
t. We show that as m tends to infinity, this conditional
probability decays to zero. The convergence is uniform in the
choice of the site x, the time t, and the configuration of the
further clusters we condition on. Being a consequence of almost
sure infinite volume convergence, we obtain uniqueness and
measurability with respect to the driving Poisson processes, and
the Markov property. The second model in focus is the Abelian
sandpile model. Let Lambda be a finite subset of the
two-dimensional integer lattice. We consider the following sandpile
model on Lambda: each vertex in Lambda contains a sandpile with a
height between one and four sand grains. At discrete times, we
choose a site v in Lambda randomly and add a sand grain at the site
v. If after adding the sand grain the height at the site v is
strictly larger than four, then the site topples. That is, four
sand grains leave the site v, and each distance-one-neighbour of v
gets one of these grains. If after toppling the site v there are
other sites with a height strictly larger than four, we continue by
toppling these sites until we obtain a configuration where all
sites have a height between one and four. We study the scaling
limit for the height one field in such a sandpile model. More
precisely, we identify the scaling limit for the covariance of
having height one at two macroscopically distant sites. We show
that this scaling limit is conformally covariant. Furthermore, we
show a central limit theorem for the sandpile height one field. Our
results are based on a representation of the height one joint
intensities that is close to a block-determinantal structure.