An Inequality for Greedy Lattice Animals

Let {X.:..Zd} be i.i.d. positive random variables with E{Xd0(log+X0)d+.}<. for some .>0 and d.2. Define Mn and Nn by Mn=max{....X.:.a self-avoiding path of lengthnstarting at the origin},Nn=max{....X.:.a lattice animal of sizencontaining the origin}. Then it has been shown that there exist M<. and N<. such that Mnn.MandNnn.Na.s. and inL1. In this paper we show that M=N if and only if X0 has bounded support and P{x0=R}.pc, where R is the right end point of support of X0 and pc is the critical probability for site percolation on Zd.