Bak-Sneppen model near zero dimension

Abstract

We consider the Bak--Sneppen model near zero dimension where the avalanche exponent $\tau$ is close to 1 and the exponents $\mu$ and $\sigma$ are close to 0. We demonstrate that $\tau -1=\mu -\sigma =\exp \{-{\mu }^{-1}-\gamma +...\}$ in this limit, where $\gamma$ 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 $\sigma$ and $\mu$ 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.

4 pages revtex (twocolumn, epsfig), 1 figure. Submitted to Phys. Rev. E, Rapid Comm