History-dependent percolation in two dimensions

Abstract

We study the history-dependent percolation in two dimensions, which evolves in generations from standard bond-percolation configurations through iteratively removing occupied bonds. Extensive simulations are performed for various generations on periodic square lattices up to side length $L=4096$. From finite-size scaling, we find that the model undergoes a continuous phase transition, which, for any finite number of generations, falls into the universality of standard 2D percolation. At the limit of infinite generation, we determine the correlation-length exponent $1/\nu=0.828(5)$ and the fractal dimension $d_{\rm f}=1.864\,4(7)$, which are not equal to $1/\nu=3/4$ and $d_{\rm f}=91/48$ for 2D percolation. Hence, the transition in the infinite-generation limit falls outside the standard percolation universality and differs from the discontinuous transition of history-dependent percolation on random networks. Further, a crossover phenomenon is observed between the two universalities in infinite and finite generations.