Percolation sliding localization and relaxation in topologically closed circuits

The glassy version of random walk on disordered lattice features a percolation related crossover to variable range hopping or to sub diffusion in one dimension The more general problem of random walk in random environment where transition rates are allowed to be asymmetric has been explored by Sinai Derrida and followers It turns out that for any small amount of disorder an unbiased spreading in one dimension becomes sub diffusive while for bias that exceeds a finite threshold there is a sliding transition leading to a non zero drift velocity In the present study we explore the implications of the percolation and sliding transitions for the relaxation modes of a topologically closed circuit A complementary question regarding the delocalization of eigenstates of non hermitian Hamiltonians has been addressed by Hatano Nelson and followers But we show that for a conservative random walk dynamics the implied spectral properties are dramatically different