Bounds for Avalanche Critical Values of the Bak - Sneppen Model

#### A. Gillett, R. Meester, M. Nuyens

2006, v.12, №4, 679-694

ABSTRACT

We study the Bak - Sneppen model on locally finite transitive graphs $G$, in particular on $\math {Z}^d$ and on $T_{\Delta}$, the regular tree with common degree $\Delta$. We show that the avalanches of the Bak - Sneppen model dominate independent site percolation, in a sense to be made precise. Since avalanches of the Bak - Sneppen model are dominated by a simple branching process, this yields upper and lower bounds for the so-called avalanche critical value $p_c^{BS}(G)$. Our main results imply that $1/ (\Delta+1) \le p_c^{BS}(T_\Delta) \le 1/(\Delta -1)$, and that $1/(2d+1) \leq p_c^{BS}(\math {Z}^d) \leq 1/2d + 1/(2d)^2 + O(d^{-3})$, as $d\to\infty$.

Keywords: Bak - Sneppen model,critical values,coupling,site percolation,branching process