The Statistics of Branched Polymers and Lattice Animals.

The properties of linear polymers, branched polymers, and percolating clusters are reviewed. Field theories that generate the statistics of linear polymers and of percolating clusters are also reviewed. Flory's approximation is used to calculate the upper critical dimension, below which mean field theory breaks down, for the above models. The exponent (upsilon), which controls the size of the polymer or percolating cluster, is calculated within Flory's approximation. The calculated value of (upsilon), in 2 and 3 dimensions, is found to be in excellent agreement with the best numerical estimates. The statistics of crosslinked polymer chains, produced by condensation of polyfunctional units, is described by constrained equilibrium ensembles with fugacities controlling dimer, trimer, endpoint and polymer number. It is shown that this description can be reproduced identically by a statistical field theory from which polymer size distribution functions can be calculated. The field theory, in the mean field approximation, in the absence of repulsive interactions, gives critical probabilities and polymer size distribution functions identical to those of Flory and Stockmayer. Modifications to the Flory-Stockmayer theory resulting from repulsive interactions are studied. Within the context of the constrained equilibrium ensembles studied here, the critical properties of gelation and percolation are identical. The field theory is analyzed in the dilute limit, i.e. on an isolated branched polymer. In this limit the critical behavior is not that of percolation. Mean field theory for this model is valid above 8 dimensions. Renormalizaton group calculations gives (epsilon)-expansions ((epsilon) = d-8) for the critical exponents. The differential recursion relations are integrated to obtain the scaling forms of the vertex functions. The hyperscaling relationship (d(upsilon) = 2-(alpha)) is found to be violated. ARB(,2) branched polymers are treated by a field theoretical approach. In the non-dilute limit the critical point is shown to be inaccessible. In the dilute limit the critical behavior is related to the critical behavior of the RA(,3) dilute limit (lattice animals).