The Continuity of M and N in Greedy Lattice Animals

Let {X_v: v ∈ Z^d}, d ≥ 2, be i.i.d. positive random variables with the common distribution F which satisfy, for some a > 0, ∫ x^d(log⁺ x)^d+a dF(x) < ∞. Define M_n = max {Σ_v∈π X_v: π a selfavoiding path of length n starting at the origin} N_n = max {Σ_v∈ξ X_v : 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 lim_{n → ∞} M_n/n and lim_{n → ∞} N_n/n = N a.s. and in L¹.