A rigorous bound on the critical exponent for the number of lattice trees,animals, and polygons

The number ofn-site lattice trees (up to translation) is believed to behave asymptotically asCn ^{–0 n} , where is a critical exponent dependent only on the dimensiond of the lattice. We present a rigorous proof that (d–1)/d for anyd2. The method also applies to lattice animals, site animals, and two-dimensional self-avoiding polygons. We also prove that v whend=2, wherev is the exponent for the radius of gyration.