Exact equations and scaling relations forf0avalanche in the Bak-Sneppen evolution model

An infinite hierarchy of exact equations is derived for the newly observed $f_0$ 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 $\rho =1\mathrm{}$ $(\rho$ is the exponent of relaxation to attractor). Scaling relations are established among the critical exponents $(\gamma ,$ $\tau ,$ D, $\sigma ,$ and $\nu )$ for the $f_0$ avalanche.