  We study several lattice random walk models with stochastic resetting to
previously visited sites which exhibit a phase transition between an anomalous
diffusive regime and a localization regime where diffusion is suppressed. The
localized phase settles above a critical resetting rate, or rate of memory use,
and the probability density asymptotically adopts in this regime a
non-equilibrium steady state similar to that of the well known problem of
diffusion with resetting to the origin. The transition occurs because of the
presence of a single impurity site where the resetting rate is lower than on
other sites, and around which the walker spontaneously localizes. Near
criticality, the localization length diverges with a critical exponent that
falls in the same class as the self-consistent theory of Anderson localization
of waves in random media. The critical dimensions are also the same in both
problems. Our study provides analytically tractable examples of localization
transitions in path-dependent, reinforced stochastic processes, which can be
also useful for understanding spatial learning by living organisms.

Statistical Mechanics;Populations and Evolution