Scaling properties of the perimeter distribution for lattice animals, percolation and compact clusters

Scaling properties of the cluster distribution function and mean cluster size in the ensemble of clusters with fixed perimeter are analysed. The relevant scaling exponents are determined in all three regions of interest: lattice animals ( p < p c ) , percolation ( p = p , ) and compact clusters ( p > p c ) . Also, a form of the lattice animal distribution function in the statistical ensemble of clusters with fixed perimeter is presented. In particular, we compare the exponents a and D, defined through (I) , so and (s), r p , where ( I ) , is the mean perimeter I of s-site clusters (the s-ensemble) and (s), is the corresponding quantity in the 1-ensemble. The percolation model has been extensively analysed during the last decade (Stauffer 1979, 1985, Essam 1980 and references therein), both because of its formal appeal and its practical importance. This model and its variants form a basis for the description of geometric aspects in a number of phenomena such as polymerisation and gelation, catalysis, hydrodynamics and cloud processes. One is typically interested in statistics of clusters formed by the process of random occupation of elements (sites or bonds) on a lattice. A cluster is a group of connected occupied elements, characterised by the number s of elements it contains (cluster elements) and the number t of its unoccupied nearest neighbours (perimeter elements). Each element is occupied with probability p and the two parameters s and t determine the statistical probability p’(1 p ) ‘ for the occurrence of the cluster. For most of the work done on percolation theory and related topics one focuses on a single parameter function n,( p ) , the distribution of clusters of mass s, which sums over all the information about the perimeter t of the clusters. On the other hand, there are some applications (e.g. colloidal catalysis) where the cluster mass is irrelevant. Catalytic activity is proportional to the number of empty sites neighbouring the cluster (‘catalytic surface’ of the cluster), which is identical to the cluster perimeter. In this case the appropriate statistical distribution should be the one which specifies the fraction of clusters having a given perimeter t , regardless of their mass s. Starting with this ‘perimeter distribution’ function one can calculate various statistical averages, which we will refer to as the r-ensemble averages, in contrast to the s-ensemble averages commonly studied. In what follows, we will show that the cluster perimeter distribution and the related averages have some interesting scaling features which have not been apparent from the analysis based on the usual s-ensemble approach. The scaling behaviour of these quantities will be analysed in all three regions of interest: ( i ) percolation ( p = p c ) , (ii) lattice animals ( p < p , ) and (iii) compact clusters ( p > p J . 0305-4470/87/090587 + 08$02.50 @ 1987 IOP Publishing Ltd L587 L588 Letter to the Editor ( a ) The s-ensemble. The average number of s-element clusters per lattice site n , ( p ) is defined as 4 ( p ) = C P ' ( l P ) ' g , r ( 1 ) f where g , , represents the number of geometrically distinct clusters with s occupied elements and t perimeter elements. In the case of the s-ensemble averages, the mean cluster perimeter as a function of q = 1 p is defined as