Greedy Lattice Animals II: Linear Growth

Let {X.:..Zd} be i.i.d. positive random variables and define Mn=max{....X.:.a self-avoiding path of lengthnstarting at the origin}, Nn=max{....X.:.a lattice animal of sizencontaining the origin}. In a preceding paper it was shown that if E{Xd0(log+X0)d+a}<. for some a>0, then there exists some constant C such that w.p.1, 0.Mn.Nn.Cn for all large n. In this part we improve this result by showing that, in fact, there exist constants M,N<. such that w.p.1, Mn/n.M and Nn/n.N.