Two Models of Magnetic Materials with Quenched Disorder

In this work I consider two models of magnetic materials with quenched disorder. The first is an XY model, on the cubic lattice Z^ d, with an additional anisotropy at each site. The directions of the anisotropy axes are taken to be independent and identically distributed random variables. This is known as the "random anisotropy model" (RAM). The second model is a Dyson-type hierarchical model for a two-dimensional Coulomb gas with quenched disorder in two forms: as fractional charges, and as a quenched array of random variables interacting with the charges via a potential of the form 1/r ^alpha, where r is the distance. The case alpha = 1 is intended to simulate a quenched array of random dipoles. For the RAM I show that if the distribution of the angles of the anisotropy axes is distributed over a cone about the x-axis of width theta_ {max}<=qpi/16, then there is a low temperature ferromagnetic phase. Further, several correlation inequalities are proven for a wider range of disorder. For the hierarchical model transformations of the single site charge activities are derived using the scaling properties of this model. These transformations are studied in a "linear" approximation. It is shown that with probability one, for alpha > 1, the non-zero charge activities converge to zero on sufficiently long length scales provided T is sufficiently small. In the case alpha = 1 this result is established only in an intermediate range of temperatures. The case alpha = 1 is analyzed using the replica method, and it is found that the non -zero charge activities converge to zero for all sufficiently low temperatures. This appears to rule out a low temperature re-entrant transition that is seen in some disordered Coulomb gas models. Finally, the annealed model is studied. At temperatures below a critical value large charges form on all length scales of the system. However, in an intermediate range of temperatures the non-zero charge activities converge to zero on sufficiently long length scales. Also, a lower bound on the fractional charge correlation function is given.