Expansion around the mean-field solution of the Bak-Sneppen model

We study a recently proposed equation for the avalanche distribution in the Bak-Sneppen model. We demonstrate that this equation indirectly relates tau, the exponent for the power law distribution of avalanche sizes, to D, the fractal dimension of an avalanche cluster. We compute this relation numerically and approximate it analytically up to the second order of expansion around the mean-held exponents. Our results are consistent with Monte Carlo simulations of the Bak-Sneppen model in one and two dimensions.