Finite Lattice Renormalization-Group Approach to Site Percolation Problem

T. Tatsumi
1977 Progress of theoretical physics  
Recently various renormalization-group approaches!) -5 ) have been proposed to study the percolation problem. 6 l The bond percolation problem has been investigated by using several methods: The relation between percolation and Ashkin-Teller-Potts model/ 1 the decimation transformations, 2 l · 3 l and the Migdal recursion relations!) Also Reynolds et al_5l studied the site and bond percolation with the cluster approach. Previously the author and Kawasaki 71 proposed the renormalization-group
more » ... ns-formation for the dilute bond Ising model by the cumulant expansion method 81 and obtained the critical concentration for the triangular lattice. In this short note we first briefly describe the results derived from the previous ap-proach7l and consider the application of Migdal-Kadanoff transformation 9 l • 101 to the site percolation problem. We consider the renormalization-group transformation 7 ) in first order for a square lattice where each cell contains five sites. The critical concentration Pc for a bond and site model are found to be 0.532 and 0.709, respectively. The result for a bond model agrees well with the known value 6 l Pc = 0.5, but for a site model the agreement with the known value 61 Pc = 0.59 is poor. Thus we need to improve the approach to the random site model. To this end, we adopt the generalized verswn of Migdal-Kadanoff transforma-
doi:10.1143/ptp.58.1056 fatcat:wy7t5gcca5gx5c2p3idtrjwpzq