Sharp Phase Boundaries for a Lattice Flux Line Model

We consider a model of nonintersecting flux lines in a rectangular region on the lattice ^d , where each flux line is a non-isotropic self-avoiding random walk constrained to begin and end on the boundary of the region. The thermodynamic limit is reached through an increasing sequence of such regions. We prove the existence of several distinct phases for this model, corresponding to different regimes for the flux line density—a phase with zero density, a collection of phases with maximal density, and at least one intermediate phase. The locations of the boundaries of these phases are determined exactly for a wide range of parameters. Our results interpolate continuously between previous results on oriented and standard nonoriented self-avoiding random walks.