Self-avoiding walks and trees in spread-out lattices

LetG_R be the graph obtained by joining all sites ofZ^d which are separated by a distance of at mostR. Let (G_R) denote the connective constant for counting the self-avoiding walks in this graph. Let (G_R) denote the coprresponding constant for counting the trees embedded inG_R. Then asR, (G_R) is asymptotic to the coordination numberk_R ofG_R, while (G_R) is asymptotic toek_R. However, ifd is 1 or 2, then (G_R)-k_R diverges to –.Dedicated to Oliver Penrose on this occasion of his 65th birthday.