Percolation, fractals and the finite-size scaling of existence probability

Based on the connection between phase transitions of thermodynamic systems and percolation transitions, we obtain the geometrical meaning of critical exponents using finite-size scaling of the percolation probability P and the existence probability E p for percolating clusters and percolating subgraphs, respectively, where E p is the ratio of the total probability weight for percolating subgraphs to the total probability weight for all subgraphs. We use a histogram Monte Carlo simulation method to calculate E p for a q -state bond-correlated percolation model (QBCPM), which is corresponding to the q -state Potts model, on the square lattice with various linear dimensions. We find that for a given q , E p has very good scaling behavior. But E p for different q has different scaling functions.