abstract:
In the
Bak-Sneppen model there are N species arranged on a circle, each of which is
assigned a random fitness. The fitnesses are independent and uniformly
distributed on (0,1). At each discrete time step the system iterates by
locating the lowest fitness and replacing this fitness, and those of its two
neighbours, by independent and uniform (0,1) random variables.
By introducing the notion of avalanche, it is possible to study Bak-Sneppen
models on all locally finite graphs. With these avalanches we can associate
a critical value. This avalanche critical value can be considered as a
hidden tuning parameter of the Bak-Sneppen model.
In the talk I hope to show that the avalanches of the Bak-Sneppen model
dominate independent site percolation, in a sense to be made precise.
Since avalanches of the Bak-Sneppen model are themselves dominated by a
simple branching process, this yields upper and lower bounds for the
avalanche critical value. These bounds are quite sharp for the regular tree
and the d-dimensional lattice as d tends to infinity.
abstract:
We study intermittency for the
Parabolic Anderson Model when the diffusion is driven by a constant $\kappa$
and the branching is induced either by an exclusion process or a voter
model. We consider the annealed Lyapunov exponents and we show that they
display an interesting dependence on $\kappa$, with qualitatively different
behaviours in different dimensions.