Tom Bannink will talk about the Bak-Sneppen process, a random process on a graph, introduced in 1993. This talk is about work done together with András Gilyén, Harry Buhrman and Mario Szegedy. Tom Bannink will explain what a phase transition is and define some functions related to this phase transition. The power series of these functions turn out to have interesting properties, which is proven in the article. He’ll explain these properties and give a short sketch of one of the proofs.
The abstract of our paper:
We consider a class of random processes on graphs that include the discrete Bak-Sneppen (DBS) process and the several versions of the contact process (CP), with a focus on the former. These processes are parametrized by a probability 0 ≤ p ≤ 1 that controls a local update rule. Numerical simulations reveal a phase transition when p goes from 0 to 1. Analytically little is known about the phase transition threshold, even for one-dimensional chains. In this article we consider a power-series approach based on representing certain quantities, such as the survival probability or the expected number of steps per site to reach the steady state, as a power-series in p. We prove that the coefficients of those power series stabilize as the length n of the chain grows. This is a phenomenon that has been used in the physics community but was not yet proven. We show that for local events A, B of which the support is a distance d apart we have cor(A, B) = O(p^d). The stabilization allows for the (exact) computation of coefficients for arbitrary large systems which can then be analyzed using the wide range of existing methods of power series analysis.