### Rigorous bounds for the discrete Bak-Sneppen model.

Probability Seminar

5th November 2021, 11:00 am – 12:00 pm

Fry Building, 2.04

The discrete version of the famous Bak-Sneppen model (https://en.wikipedia.org/wiki/Bak-Sneppen_model) is a Markov chain on the space of {0,1} sequences of length n with periodic boundary conditions, which runs as follows. Fix some p. At each moment of (discrete) time, one of the zeros is chosen at random with equal probability. Then this zero and both its neighbours are replaced by three independent Bernoulli(p) random variables. This procedure is repeated ad infinitum.

Barbay and Kenyon (2001) claimed that the fraction of zeros in the stationary distribution becomes negligible when n goes to infinity whenever p>0.54. This result is indeed correct, however, its proof is not.

I shall present the rigorous proof of the Barbay and Kenyon's result, as well as some better bounds for the critical p.

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