Z. Phys. B - Condensed Matter 45, 123-128 (1981) Condensed Zeitschrift Matter fiJr Physik B 9 Springer-Verlag 1981 Radius of Clusters at the Percolation Threshold: A Position Space Renormalization Group Study Fereydoon Family* and Peter J. Reynolds** Center for Polymer Studies*** and Department of Physics, Boston University, Boston, Massachusetts, USA Received August 17, 1981 Using a direct position-space renormalization-group approach we study percolation clusters in the limit s ~ o% where s is the number of occupied elements in a cluster. We do this by assigning a fugacity K per cluster element; as K approaches a critical value Kc, the conjugate variable s~oe. All exponents along the path (K-Kc)--,O are then related to a corresponding exponent along the path s ~ oe. We calculate the exponent p, which describes how the radius of an s-site cluster grows with s at the percolation threshold, in dimensions d=2,3. In d=2 our numerical estimate of p=0.52_+0.02, obtained from extrapolation and from cell-to-cell transformation procedures, is in agreement with the best known estimates. We combine this result with previous PSRG calculations for the connectedness-length exponent v, to make an indirect test of cluster-radius scaling by calculating the scaling function exponent o- using the relation o- = p/v. Our result for a is in agreement with direct Monte-Carlo calculations of a, and thus supports the cluster-radius scaling assumption. We also calculate p in d=3 for both site and bond percolation, using a cell of linear size b=2 on the simple-cubic lattice. Although the result of such small-cell calculations are at best only approximate, they nevertheless are consistent with the most recent numerical estimates. 1. Introduction The study of cluster properties in physical systems is of interest in numerous areas (such as nucleation, phase transitions, colloidal and aerosol suspensions, and polymers). A simple model of cluster properties is provided by percolation [1, 2] which has wide appeal because of its simplicity and its applicability to diverse physical phenomena. For example, perco- lation has been used as a model for gelation [3], and conduction in disordered materials [4], as well as for the study of the anomalous properties of water at low temperatures [5]. In addition, percola- tion exhibits all the complexities of systems near a second-order phase transition, and therefore pro- vides a testing ground for theories of critical-point phenomena. * Permanent address: Department of Physics, Emory University, Atlanta, GA 30322, USA ** Also at NRCC, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA *** Supported in part by grants from ARO and ONR Thus scaling and the renormalization group, which are two important tools of the modern theory of critical phenomena, have been applied to the perco- lation problem [see e.g., 6-10, and references there- in]. In addition, Nakanishi and Stanley have made exhaustive studies of the scaling of the percolation equation of state and the cluster numbers [11] in various dimensions. Percolation critical exponents have been calculated by position-space renormaliza- tion group (PSRG) [see e.g., 9, and references there- in] and e-expansions in 6-e dimensions [see e.g., 10, and references therein]. Cluster-number scaling has also been studied by PSRG [12]. However, one important scaling theory - cluster-radius scaling [8] - has thus far been studied mainly by numerical methods [2]. Since the statistics of percolation clus- ters is closely related to that of branched polymers [13-16], a study of the dependence of the percola- tion cluster radius on its size - which is the basis of cluster-radius scaling - is of particular interest in the 0340-224X/81/0045/0123/$01.20