### abstract

- The original version of Einstein’s Brownian motion problem is essentially equivalent to the analysis of a simple random walk. The more complicated version of a random walk on a disordered lattice, features a percolation-related crossover to variable-range-hopping, or to sub-diffusion in one-dimension1. In fact it is formally like a resistor-network problem, and has diverse applications, e.g. in the context of “glassy” electron dynamics2,3. But more generally one has to consider Sinai’s spreading problem4,5,6,7, aka a random walk in a random environment, where the transition rates are allowed to be asymmetric. 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. The latter has relevance e.g. for studies in a biophysical context: population biology8,9, pulling pinned polymers and DNA denaturation10,11 and processive molecular motors12,13. The dynamics in all the above variations of the random-walk problem can be regarded as a stochastic process in which a particle hops from site to site. The rate equation for the site occupation probabilities p = {pn} can be written in matrix notation as involving a matrix W whose off-diagonal elements are the transition rates wnm, and with diagonal elements −γn such that each column sums to zero. Assuming near-neighbor hopping the W matrix takes the form In Einstein’s theory W is symmetric, and all the non-zero rates are the same. Contrary to that, in the “glassy” resistor-network problem (see Methods) the rates have some distribution P(w) whose small w asymptotics is characterized by an exponent α, namely P(w) ∝ wα−1 for small w. To be specific we consider The conductivity of the network w∞ is sensitive to α. It is given by the harmonic average over the wn, reflecting serial addition of connectors. It comes out non-zero in the percolating regime (α > 1). For the above distribution w∞ = [(α − 1)/α]wc. In Sinai’s spreading problem W is allowed to be asymmetric. Accordingly the rates at the nth bond can be written as for forward and backward transitions respectively. For the purpose of presentation we assume that the stochastic field is box distributed within [s − σ, s + σ]. We refer to s as the bias: it is the pulling force in the case of depinning polymers and DNA denaturation; or the convective flow of bacteria relative to the nutrients in the case of population biology; or the affinity of the chemical cycle in the case of molecular motors. Our interest is in the relaxation dynamics of finite N-site ring-shaped circuits14,15, that are described by the stochastic equation equation (1). The ring is characterized by its so-called affinity, The N sites might be physical locations in some lattice structure, or can represent steps of some chemical-cycle. For example, in the Brownian motor context N is the number of chemical-reactions required to advance the motor one pace. We are inspired by the study of of non-Hermitian quantum Hamiltonians with regard to vortex depinning in type II superconductors16,17,18; molecular motors with finite processivity19,20; and related works21,22,23. In the first example the bias is the applied transverse magnetic field; and N is the number of defects to which the magnetic vortex can pin. In both examples conservation of probability is violated.