PART I: QUANTUM FLUCTUATIONS IN CHAINS OF JOSEPHSON JUNCTIONS. PART II: DIRECTED AGGREGATION ON THE BETHE LATTICE

This thesis is divided into two distinct parts. In Part One, we study the effect of quantum fluctuations of the phase on the low-temperature behavior of two models of Josephson junction chains with Coulomb interactions taken into account. The first model, which represents a chain of junctions close to a ground plane, is the Hamiltonian version of the two-dimensional XY model in one space and one time dimension. We demonstrate explicitly how the Nelson-Kosterlitz jump manifests itself in the conduction properties of this system at a critical value of the superconducting grain capacitance. In the second model, the charging energy for a single junction in the chain is just the parallel-plate capacitor energy. We show that quantum fluctuations produce exponential decay of the order-parameter correlation junction for any finite value of the junction capacitance. Therefore, in contrast to the first model, the Coulomb interaction always succeeds in disrupting the phase coherence of the array.; Part Two deals with two types of directed aggregation on the Bethe lattice--directed diffusion-limited aggregation (DLA) and ballistic aggregation (BA). In the DDLA problem on finite lattices, we construct an exact nonlinear recursion relation for the probability distribution of the density. The mean density tends to zero as the lattice size is taken to infinity. Using a mapping between the model with perfect adhesion on contact and another model with a particular value of the adhesion probability, we show that the adhesion probability is irrelevant over an interval of values. The aggregates in a problem on the infinite lattice display marginal scaling behavior: they are compact up to a logarithmic correction for any coordination number of the lattice. In the BA problem we calculate the mean level number and consider fluctuation effects. The implications of a mapping from the BA problem to the DDLA problem on finite lattices are discussed.