Super-scaling of Percolation on Rectangular Domains

For percolation on a $(RL) \times L$ two-dimensional rectangular domains with width $L$ and aspect ratio $R$, we propose that the existence probability of percolating cluster $E_p(L, \epsilon, R) $ as a function of $L$, $R$, and deviation from the critical point $\epsilon$ can be expressed as $F(\epsilon L^{y_t}R^a)$, where $y_t\equiv 1/\nu$ is the thermal scaling power, $a$ is a new exponent, and $F$ is a scaling function. We use Monte Carlo simulation of bond percolation on square lattices to test our proposal and find that it is well satisfied with $a=0.14(1)$ for $R > 2$. We also propose super-scaling for other critical quantities.