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).