Bounds for criticals value of the Bak-Sneppen model on transitive graphs

with Alexis Gillett and Ronald Meester

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Abstract: We study the Bak-Sneppen model on locally finite transitive graphs G, in particular on Z^d and on T_D, the regular tree with common degree D. We show that the avalanches of the Bak-Sneppen model dominate independent site percolation, in a sense to be made precise. Together with the fact that avalanches of the Bak-Sneppen model are dominated by a simple branching process, this yields upper and lower bounds for the critical value p_c^{BS}(G) of the Bak-Sneppen model. Our main results state that 1/(Delta+1)<= p_c^{BS}(T_\Delta) <= 1/(\Delta -1), and that 1/(2d+1)<= p_c^{BS}(\mathbb{Z}^d)<=1/(2d)+ 1/(2d)^2+O(d^{-3}), as d\to\infty.

appeared in Markov processes and related fields 62:679:694 (2006)