Transition to localization of biased walkers in a randomly absorbing environment

We study biased random walkers on lattices with randomly dispersed static traps in one, two and three dimensions. As the external bias is increased from zero the system undergoes a phase transition, most clearly manifested in the asymptotic drift velocity of survivors which is zero below a critical bias. This transition is second order in one dimension but of first order in higher dimensions. The model can be mapped to a stretched polymer with attractive interaction between monomers, and this phase transition would then describe sudden unfolding of the polymer when the stretching force exceeds a critical value. We also present precise simulations of the zero-bias case where we show unambiguously that the transition between the Rosenstock (RS) and Donsker–Varadhan (DV) regimes is first order in dimension ≥2.