Exact solution of the one-dimensional percolation problem with further neighbour bonds?

We obtain an exact solution for the one-dimensional site percolation model with bonds connecting the Lth nearest neighbours. The critical exponents are found to depend on L: 2 a p = L, yp = L, vP = L, qp = 1, pp = 0 and 6, = 00. By mapping the percolation problem onto an king-like model with multi-spin interactions, we argue that the dependence of the percolation exponents on L can be understood from consideration of king model universality classes. The percolation problem (Shante and Kirkpatrick 197 1) has attracted considerable interest in recent years. The concepts of scaling and universality, which are so useful in understanding the behaviour at the critical point, have also proved to be important in the investigation of the percolation transition. In this Letter we solve a one-dimensional (d = 1) site percolation problem with the addition of further neighbour bonds. The generating function (Fisher and Essam 1961) which is the mean number of clusters per site, is where (n,) is the average number of s-site clusters per lattice site, and p is the probability that a site is occupied. The prime on the sum indicates that the infinite cluster is omitted. First we introduce a variable that plays a role analogous to the magnetic field in the thermal problem (Kasteleyn and Fortuin 1969). To each site in the lattice we associate a ‘ghost site’][ which is occupied with probability h (cf figure 1). All the ghost sites are connected to one another, so that if a site on the lattice is occupied, it is part of the infinite cluster if its corresponding ghost site is occupied (Reynolds et a1 1977). Hence, (n,) = PS(4)(1 hY1pS, (2) where D,(q) is the perimeter polynomial for h = 0, and q = 1 p . The factor (1 h)” in the generalised (h ZO) perimeter polynomial ensures that the s-site cluster is isolated. t Work supported by AFOSR and NSF. $ Present address: Physics Department, University of Toronto. 8 Predoctoral Fellow, Massachusetts Institute of Technology. 11 We could also define a single ghost spin connected by ghost bonds to each site and obtain an identical formalism. See also Reynolds et al (1977).