Exact equations and scaling relations for f0 avalanche in the Bak-Sneppen evolution model

An infinite hierarchy of exact equations is derived for the newly observed (f) over bar(0) avalanche in the Bak-Sneppen model. By solving the first-order exact equation, we find that the critical exponent gamma, governing the divergence of the average avalanche size, is exactly 1 (for all dimensions), which has been confirmed by extensive simulations. Solution of the gap equation yields another universal result rho = 1 (rho is the exponent of relaxation to attractor). Scaling relations are established among the critical exponents (gamma, tau, D, sigma, and nu) for the (f) over bar(0) avalanche.