# SELF-AVOIDING WALKS AND TREES IN SPREAD-OUT LATTICES

## by Mathew D. Penrose

Let $\G_R$ be the graph obtained by joining all sites of $Z^d$ which are separated by a distance of at most $R$. Let $\mu(\G_R)$ denote the connective constant for counting the self-avoiding walks in this graph. Let $\lambda(\G_R)$ denote the corresponding constant for counting the trees embedded in $\G_R$. Then as $R$ goes to infinity, $\mu(\G_R)$ is asymptotic to the co-ordination number $k_R$ of $\G_R$, while $\lambda(\G_R)$ is asymptotic to $e k_R$. However, if $d$ is 1 or 2, then $\mu(\G_R) - k_R$ diverges to $-\infty$.

Journal of Statistical Physics 77, 3-15 (1994).