Localization induced by Sinai disorder in mass-exchange models

Abstract

We introduce a conservative continuous-mass-exchange model with quenched disorder in the splitting laws on the bonds of a finite segment. Although the dynamics is generically irreversible, its stationary mean mass profile remarkably turns out to be exactly solvable through an associated birth-and-death chain and has a Gibbs representation in terms of a simple random-walk potential. If $N$ is the number of bonds and $β$ measures the strength of disorder, the parameter $β\sqrt N$ separates an asymptotically flat profile, a Brownian crossover, and a profile localized near the global minimum. We characterize the fluctuations of macroscopic masses and identify arcsine laws for the localization centre, the competition between left and right valleys, and the disorder-averaged mass measure. Furthermore, for the extremal masses, we discuss the emergent limiting random variable in terms of the literature linking finite RW paths seen from its minimum to infinite-horizon walks conditioned to stay positive. Finally, in proximity of the localization transition point $β=0$, we show that the inverse participation ratio for the extremal masses give rises to a scaling with decay of order $k^{-3/2}$ consistent with the scaling limit of the latter conditioned walk. In the same limit, the inverse participation ratio shows a scaling form.