N. Madras, A RIGOROUS BOUND ON THE CRITICAL EXPONENT FOR THE NUMBER OF LATTICE TREES, ANIMALS, AND POLYGONS, Journal of statistical physics, 78(3-4), 1995, pp. 681-699
The number of n-site lattice trees (up to translation) is believed to
behave asymptotically as Cn(-0)lambda(n), where theta is a critical ex
ponent dependent only on the dimension d of the lattice. We present a
rigorous proof that theta greater than or equal to (d-1)/d for any d g
reater than or equal to 2. The method also applies to lattice animals,
site animals, and two-dimensional self-avoiding polygons. We also pro
ve that theta greater than or equal to nu when d = 2, where nu is the
exponent for the radius of gyration.