Asymptotic function for multi-growth surfaces using power-law noise

Numerical simulations are used to investigate the multiaffine exponent $\alpha_q$ and multi-growth exponent $\beta_q$ of ballistic deposition growth for noise obeying a power-law distribution. The simulated values of $\beta_q$ are compared with the asymptotic function $\beta_q = \frac{1}{q}$ that is approximated from the power-law behavior of the distribution of height differences over time. They are in good agreement for large $q$. The simulated $\alpha_q$ is found in the range $\frac{1}{q} \leq \alpha_q \leq \frac{2}{q+1}$. This implies that large rare events tend to break the KPZ universality scaling-law at higher order $q$.