Transfer-Matrix Study of the Elastic Properties of Central-Force Percolation

Abstract

We study two versions of central-force percolation on a triangular lattice made out of springs that can freely rotate around nodes: 1) random dilution; a proportion 1 - p of springs is missing and 2) random reinforcement; a proportion p of springs is infinitely rigid, while the other ones are all present and have finite strength. We use a transfer-matrix algorithm to deal with strips of width ranging from 2 to 30 in case 1) and 48 in case 2), and length 10⁵. We estimate the threshold p* = 0.642 and we obtain the critical exponents relative to the scaling of the elastic moduli with the strip width in case 1) 3.0 ± 0.4 and 2) 0.97 ± 0.02. These values are very close to the ones found for percolation systems with angular elasticity, supporting the hypothesis that these two critical phenomena belong to the same universality class.