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

TL;DR

This paper introduces a simple method to accurately determine thresholds and critical exponents in Bak-Sneppen models, analyzing different variants and their universality classes through extensive simulations.

Contribution

The study provides precise threshold and exponent measurements for various Bak-Sneppen models and explores the universality classes based on symmetry considerations.

Findings

Thresholds match theoretical predictions for different models.

Critical exponents vary with model variants, indicating different universality classes.

Models exhibit singular stationary probability distributions.

Abstract

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.