Two-dimensional percolation model with long-range interaction

We perform large-scale simulations of the two-dimensional long-range bond percolation model with algebraically decaying percolation probabilities $\sim 1\mathrm{/}{r}^{2+\sigma }$, using both conventional ensemble and event-based ensemble methods for system sizes up to $L=16384$. We accurately determine the critical points, the universal values of several dimensionless quantities, and the corresponding critical exponents. Our results provide compelling evidence that the system undergoes a crossover from short-range to long-range universality at $\sigma =2$, in contradiction to Sak’s criterion. Notably, we observe a pronounced jump in the universal values and critical exponents at $\sigma =2$, a feature absent from previous studies.