The Continuity of M and N in Greedy Lattice Animals

Abstract

Let {Xv: v ∈ Zd}, d ≥ 2, be i.i.d. positive random variables with the common distribution F which satisfy, for some a > 0, ∫ xd(log+ x)d+a dF(x) < ∞. Define Mn = max {Σv∈π Xv: π a selfavoiding path of length n starting at the origin} Nn = max {Σv∈ξ Xv : ξ a lattice animal of size n containing the origin} Then it has been shown that there exist positive finite constants M = M[F] and N = N[F] such that M N limn → ∞ Mn/n and limn → ∞ Nn/n = N a.s. and in L1.