Bak-Sneppen model near zero dimension

Abstract

We consider the Bak--Sneppen model near zero dimension where the avalanche exponent $τ$ is close to 1 and the exponents $μ$ and $σ$ are close to 0. We demonstrate that $τ-1=μ-σ=\exp\{-μ^{-1}-γ+...\}$ in this limit, where $γ$ is the Euler's constant. The avalanche hierarchy equation is rewritten in a form that makes possible to find the relation between the critical exponents $σ$ and $μ$ with high accuracy. We obtain new, more precise, values of the critical exponents for the 1D and 2D Bak-Sneppen model and for the 1D anisotropic Bak-Sneppen model.