Localization-delocalization transition in low-dimensional asymmetric resetting random walks

We investigate biased random walks with stochastic resetting to previously visited sites and demonstrate how directional drift fundamentally alters the emergence of localized stationary states. We show that bias acts as a competing mechanism and can weaken or suppress localization. This competition produces an anisotropic phase transition: while the walker remains localized transverse to the bias, the localization length diverges only along the direction of the drift, independently of spatial dimension. We derive the stationary distribution, identify the conditions under which localization survives in the presence of bias, and compute the critical behavior of the localization length near the transition. These results reveal how the interplay of bias and resetting reshapes the phase structure of nonequilibrium random walks and establishes directional divergence as a hallmark of biased resetting dynamics.