A real-space renormalization group for site and bond percolation?

We develop a real-space renormalization group which renormalizes probabilities directly in the percolation problem. An exact transformation is given in one dimension, and a cluster approach is presented for other lattices. Our results are in excellent agreement with series calculations for the critical percolation concentration p, (both site and bond), and in good agreement for the correlation length exponent vp. Additionally, in one dimension we include a field-like variable and calculate the remaining exponents. The percolation problem (Shante and Kirkpatrick 1971) has been receiving renewed attention lately, and with its similarity to thermal phase transitions, the renormalization group (RG) comes readily to mind. See, for example, Wilson and Kogut (1974) and Niemeyer and Van Leeuwen (1974). Several workers have in fact attempted various RG approaches, and these have met with fair success. Harris et a1 (1975) and Dasgupta (1976) used the fact that the bond percolation problem has a direct mapping onto the s-state Ashkin-Teller-Potts model when s -P 1 (Kasteleyn and Fortuin 1969). They then use this model to do RG transformations both in real space and in eexpansion from d = 6 dimensions. Others have transformed the bond probabilities directly. In particular, Young and Stinchcombe (1975) and Stinchcombe and Watson (1976) use decimation, while Kirkpatrick (1 977) uses Migdal recursion relations. The previous work concerns only the bond percolation problem. In this Letter we present a cluster approach which works on the probabilities directly, and applies to site as well as bond percolation. First we treat the site problem. We start by choosing a lattice which we partition into cells that both cover the lattice and maintain its original symmetry (figure 1). These cells will play the role of renormalized sites. Given that sites in the original lattice are independently occupied with probability p , we must choose a cell occupation probability p’ = W(p) in such a way that W(p) contains the essential physics of percolation. To this end we note that percolation involves the formation of an infinite connected network-that is, one that actually ‘gets across’ the entire lattice. Below percolation only finite clusters are present. Thus it is sensible to define a cell as occupied if and only if it contains a set of sites such that the cell ‘percolates.’ This will define %?@) for us. Since our transformation rescales the lattice spacing by a factor of b, we expect the t Work supported by NSF and AFOSR