SELF-AVOIDING WALKS AND TREES IN SPREAD-OUT LATTICES

Let G(R) be the graph obtained by joining all sites of Z(d) which are separated by a distance of at most R. Let mu(G(R)) denote the connective constant for counting the self-avoiding walks in this graph. Let lambda(G(R)) denote the corresponding constant for counting the trees embedded in G(R). Then as R --> infinity, mu(G(R)) is asymptotic to the coordination number k(R) of G(R), while lambda(G(R)) is asymptotic to ek(R). However, if d is 1 or 2, then mu(G(R)) - k(R) diverges to - infinity.