abstract:
In the sandpile model on Z^d, we start with an initial configuration \eta that consists of a number of sand grains on each lattice site, and then topple each unstable site, that is, each site that contains at least 2d grains. In a toppling, the site gives one grain to each neighbor. The toppling process may or may not yield a final configuration with only stable sites (in that case, we call \eta stabilizable), and if it does, we may or may not have percolation of sites that toppled. The occurrence of these possibilities depends only on \eta, and not on the particular order in which we perform the topplings, provided some mild conditions are satisfied.
As a parameter, we use the density \rho of sand grains, choosing \eta according to a stationary, ergodic measure \mu. The existence of phase transitions, and thus of critical value(s) for \rho, is an open problem for all d>1.
So far, we obtained the following
results: there exist \rho>0 such that there is no percolation of toppled
sites, and if there is a \rho_c for stabilizability, then it is between d
and 2d-1. for d=1, \rho_c=1 for both phase transitions, and if \mu is a
product measure, then the phase transition is first order.
abstract:
The Bak-Sneppen model originated as a
model of biological evolution and can be defined mathematically as follows.
One takes a graph and assigns a value between 0 and 1, called a fitness, to
each vertex. These fitnesses change in discrete time according to the
following rule. At each time step, the minimal fitness is located and this
fitness and all the fitnesses of neighbouring vertices are replaced with
independent uniform [0,1] distributed fitnesses.
There has been much interest in this
model from the physics community, because despite the simplicity of its
rules, it enjoys complex behaviour. This talk will focus on introducing the
techniques that have been used to analyse the model and on summarising the
rigorously established results about the model. There are many open problems
for this model and typically these are analogous to questions that have
posed about more well-known models such as percolation. These open problems
will be stated together with recent results.