Self-avoiding Walks In the Presence of Strongly Correlated, Annealed Vacancies

Two models coupling self-avoiding walks (SAWs) to Ising vacancies are studied. When the walk is performed on the hexagonal lattice and the vacancies are on the dual, an exact description of the behavior of the walk as a function of the Ising coupling is given. The walk collapses from SAW to FTHETA’-point behavior at the Ising critical point where its fractal dimension equals 11/8, as for the hull of Ising clusters. A walk on the square lattice, with vacancies either on the same or on the dual lattice, has SAW behavior even at Ising criticality. This is shown numerically and, within the conformal-invariance approach, poses a problem for which a conjectural solution is proposed.