On the thresholds, probability densities, and critical exponents of Bak-Sneppen-like models

We report a simple method to accurately determine the threshold and the exponent $\nu$ of the Bak-Sneppen model and also investigate the BS universality class. For the random-neighbor version of the BS model, we find the threshold $x^*=0.33332(3)$, in agreement with the exact result $x^*=1/3$ given by mean-field theory. For the one-dimensional original model, we find $x^*=0.6672(2)$ in good agreement with the results reported in the literature; for the anisotropic BS model we obtain $x^*=0.7240(1)$. We study the finite size effect $x^*(L)-x^*(L \to \infty) \propto L^{-\nu}$, observed in a system with $L$ sites, and find $\nu = 1.00(1)$ for the random-neighbor version, $\nu = 1.40(1)$ for the original model, and $\nu=1.58(1)$ for the anisotropic case. Finally, we discuss the effect of defining the extremal site as the one which minimizes a general function $f(x)$, instead of simply $f(x)=x$ as in the original updating rule. We emphasize that models with extremal dynamics have singular stationary probability distributions $p(x)$. Our simulations indicate the existence of two symmetry-based universality classes.