Shape of Random Walk and Size of Self-Avoiding Manifold.

This thesis studies the anisotropy of random walks and the size of an n-dimensional manifold with interactions. In particular, we look at a free self-avoiding manifold and one that is immersed in a medium consisting of randomly distributed impurities, and a polymer (n = 1) confined to a region between two infinite, parallel, repelling plates. The anisotropy is characterized by the quantity Delta = < tr{ bf Q}^2/(tr{bf Q})^2 >, averaged over an ensemble of random walks, where Q is the 'shape tensor', and {bf Q} = {bf Q} - {1 over d} {rm tr}{bf Q}. The approximation Delta _{o}equiv< tr{ bf Q}^2>/<(tr{ bf Q})^2> is frequently used in the literature. This differs significantly from Delta. In this thesis we use the integral representation 1/({rm tr} {bf Q})^2 = int_sp{0 }{infty} dx x exp( {-x} {rm tr}{bf Q}) together with Gegenbauer polynomials to calculate Delta. We find that the asphericity parameter has a minimum near d = 4, implying that the random walk is more spherical in intermediate dimensions than in low or high dimensions. Treating y trQ (informally) as a free energy provides a bridge connecting polymers with surfaces (n = 2), examples of which include the inner layer of red blood cells, plaquette ensembles in lattice gauge theories, and interfaces. A particular (continuous) model of a surface, and its generalization, the (n>2)-dimensional manifold, is based on Edwards' polymer model. Renormalization group (RG) calculations, in the literature, of the size exponent, nu, for such a surface are patently incorrect, and are ambiguous when specialized to two-dimensional polymers (n = 1). We analyze this model using the "effective step length" method. We find that manifolds are more extended than RG indicates. A tethered membrane in an annealed random environment undergoes a transition from an extended state, with size exponent nu = 4/5, to a 'crumpled' state, when the density of the impurities approaches a certain critical value. This is consistent with the behavior of polymers reported previously in the literature. For a polymer in a slit of width D, we find that the size scales as R~ N^{3/4 }D^{-1/4} as D/R_ {F}to0, R_{F} the Flory radius of the chain. For D/R_ {F}toinfty, we recover the three-dimensional result, R~ N^ {3/5}..