We show how certain properties of the Anderson model on a tree are related to the solutions of a non-linear integral equation. Whether the wave function is extended or localized, for example, corresponds to whether or not the equation has a complex solution.We show how the equation can be solved in a weak disorder expansion. We find that, for small disorder strength λ, there is an energy E c (λ) above which the density of states and the conducting properties vanish to all orders in perturbation theory. We compute perturbatively the position of the line E c (λ) which begins, in the limit of zero disorder, at the band edge of the pure system. Inside the band of the pure system the density of states and conducting properties can be computed perturbatively. This expansion breaks down near E c (λ) because of small denominators. We show how it can be resummed by choosing the appropriate scaling of the energy. For energies greater than E c (λ) we show that non-perturbative effects contribute to the density of states but have been unable tell whether they also contribute to the conducting properties. Short Title: Anderson Model on a Cayley tree PACS No: 71.30 71.50 J

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We argue that the field theory that descibes randomly branched polymers is not generally conformally invariant in two dimensions at its critical point. In particular, we show (i) that the most natural formulation of conformal invariance for randomly branched polymers leads to inconsistencies; (ii) that the free field theory obtained by setting the potential equal to zero in the branched polymer field theory is not even classically conformally invariant; and (iii) that numerical enumerations of the exponent θ(α),

We use the supersymmetric formulation of the branched polymer problem to calculate the pair correlation function GN(T) of a randomly branched polymer in three dimensions:Here N is the number of monomers in the polymer and RN is the polymer's radius of gyration. We also compare the Fourier transform of GN(T), the structure factor S N ( ~) ,with the results of a recent experiment, and speculate on why our result disagrees with it.

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