Abstract. Exact values of the numbers of connected clusters of n sites, each site having valence no larger than nu , are presented for the simple cubic lattice for nu =2, 3, 4, 5 and 6 for small values of n. Assuming a plausible asymptotic form for the dependence of these numbers on n and nu the authors show rigorously that the exponent tau characterising the dominant singularity in the generating function is negative for nu =2 but positive for nu >or=3. Series analysis techniques suggest that the same value of tau obtains for all nu >or=3.

Print publication: Issue 5 (1 May 1980)