Greedy lattice animals. II. Linear growth

Let {X(v): v in Z^d} be an i.i.d. family of positive random variables and define M(n) = max{ Sum X(v) for v in q: q is a self-avoiding path of length n starting at the origin}, N(n) = max{ Sum X(v) for v in D: D is a lattice animal of size n containing the origin}. In a preceding paper it was shown that if E{X^d(log(X)^(d+ a) } 0, then there exists some constant C such that w.p.1, 0 < M(n )< N(n) < Cn for all large n. In this part we improve this result by showing that, in fact, there exist finite constants M, N such that w.p.1, M(n)/n tends to M and N(n)/n tends to N.