The Continuity of M and N in Greedy Lattice Animals

Abstract

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

$$M_n = \max \left\{ {\sum\limits_{\upsilon \in \pi } {X_\upsilon } {\kern 1pt} :\pi {\text{ a selfavoiding path of length }}n{\text{ starting at the origin}}} \right\}$$

$$N_n = \max \left\{ {\sum\limits_{\upsilon \in \xi } {X_\upsilon } {\kern 1pt} :\xi {\text{ a lattice animal of size }}n{\text{ containing the origin}}} \right\}$$

Then it has been shown that there exist positive finite constants M = M[F] and N = N[F] such that

$${\mathop {\lim }\limits_{n \to \infty }} \frac{{M_n }}{n} = M{\text{ and }}{\mathop {\lim }\limits_{n \to \infty }} \frac{{N_n }}{n} = N{\text{ a}}{\text{.s}}{\text{. and in }}L^1 $$