Cluster analysis and finite-size scaling for Ising spin systems

Based on the connection between the Ising model and a correlated percolation model, we calculate the distribution function for the fraction $\mathrm{}\left(c\right)\mathrm{}$ of lattice sites in percolating clusters in subgraphs with n percolating clusters, $f_n\mathrm{}\left(c\right),$ and the distribution function for magnetization $\mathrm{}\left(m\right)\mathrm{}$ in subgraphs with n percolating clusters, $p_n\mathrm{}\left(m\right).$ We find that $f_n\mathrm{}\left(c\right)\mathrm{}$ and $p_n\mathrm{}\left(m\right)\mathrm{}$ have very good finite-size scaling behavior and that they have universal finite-size scaling functions for the model on square, plane triangular, and honeycomb lattices when aspect ratios of these lattices have the proportions $1:\sqrt{3}/2:\sqrt{3}.$ The complex structure of the magnetization distribution function $p\left(m\right)\mathrm{}$ for the system with large aspect ratio could be understood from the independent orientations of two or more percolation clusters in such a system.