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 0 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.
Rigorous bounds for the discrete Bak-Sneppen model
Tue, Jan. 25 1pm (Zoom)
Grad Algebra Seminar
Charlotte Aten (University of Rochester)
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In my recent work with Semin Yoo we produced a generalization of a construction of Herman and Pakianathan which assigns to each finite noncommutative group a closed surface in a functorial manner. We give a pair of functors whose domain is a subcategory of a variety of n-ary quasigroups. The first of these functors assigns to each such quasigroup a smooth, flat Riemannian manifold while the second assigns to each quasigroup a topological manifold which is a subspace of the metric completion of the aforementioned Riemannian manifold. I will give examples of these constructions, draw some pictures, and argue that all homeomorphism classes of smooth orientable manifolds arise from this construction. I will then discuss a connection with the Evans Conjecture on partial Latin squares, give its implication for orientable surfaces, and state a related problem applicable to our construction for compact n-manifolds.
Orientable smooth manifolds are essentially quasigroups
Rigorous bounds for the discrete Bak-Sneppen model