Transfer-matrix approach to the one-dimensional percolation problem

Abstract

The transfer-matrix method is used to find the exact phase diagrams and the correlation length exponents v for the one-dimensional site, bond and site-bond percolation problems with bonds connecting the Lth-nearest neighbours (L up to 3). For the site percolation, the authors' results agree with the exact result obtained from the generating function method, v=L, while for the bond percolation they found much richer critical phenomena. If all the L bonds have equal occupation probability, their results predict v=L(L+1)/2.