Sharp phase boundaries for a lattice flux line model

We consider a model of noninteresting flux lines in a rectangular region on the lattice Z(d), where each flux line is a non-isotropic self-avoiding ra ndom walk constrained to begin and end on the boundary of the region. The t hermodynamic limit is reached through an increasing sequence of such region s. We prove the existence of several distinct phases for this model, corres ponding to different regimes for the flux line density-a phase with zero de nsity, a collection of phases with maximal density, and at least one interm ediate phase. The locations of the boundaries of these phases are determine d exactly for a wide range of parameters. Our results interpolate continuou sly between previous results on oriented and standard nonoriented self-avoi ding random walks.