In the original Bak-Sneppen model, there are N species arranged on a circle, each having a random fitness, independently and uniformly distributed on (0,1). At every time step the system is updated by locating the smallest fitness and replacing this fitness and those of its two neighbours by independent uniform(0,1) random variables.
The Bak-Sneppen model can be broken down into a series of so-called avalanches. A p -avalanche is said to occur between times s and s + t if at time s all the fitnesses are greater than p, and time s + t is the next time after s at which this occurs.
Unlike the original model, Bak-Sneppen avalanches can be constructed on all locally finite graphs, like trees and square lattices. For avalanches on a locally finite graph G, let r(p) denote the number of vertices updated by a p -avalanche. We then define the critical value pc := inf { p > 0 : P( r(p) = ∞ ) > 0 }.
In the talk, we compare pc with the critical value of two well-known processes: pc is larger than the critical value of a certain Galton-Watson branching process, but smaller than the critical value of site percolation on G. We then apply this result to derive bounds for pc on trees and the d-dimensional square lattice.
This is joint work with Alexis Gillett and Ronald Meester (VU).