Sharp Phase Boundaries for a Lattice Flux Line Model

We consider a model of nonintersecting flux lines on the lattice Z d , where each flux line is a non-isotropic self-avoiding random walk. The thermodynamic limit is reached through an increasing sequence of rectangular regions, with flux lines constrained to begin and end on the boundary of the region. 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 non-oriented self-avoiding random walks.