The number and size of branched polymers in high dimensions

We consider two models of branched polymers (lattice trees) on the d-dimensional hypercubic lattice: (i)the nearest-neighbor model in sufficiently high dimensions, and (ii) a "spread-out" or long-range model for d>8, in which trees are constructed from bonds of length less than or equal to a large parameter L. We prove that for either model the critical exponent θ for the number of branched polymers exists and equals 5/2, and that the critical exponent v for the radius of gyration exists and equals 1/4. This improves our earlier results for the corresponding generating functions. The proof uses the lace expansion, together with an analysis involving fractional derivatives which has been applied previously to the self-avoiding walk in a similar context.