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We study in detail a one-dimensional lattice model of a continuum, conservedfield (mass) that is transferred deterministically between neighbouring randomsites. The model falls in a wider class of lattice models capturing the jointeffect of random advection and diffusion and encompassing as specific cases,some models studied in the literature, like the Kang-Redner,Kipnis-Marchioro-Presutti, Takayasu-Taguchi, etc. The motivation for our setupcomes from a straightforward interpretation as advection of particles inone-dimensional turbulence, but it is also related to a problem ofsynchronization of dynamical systems driven by common noise. For finitelattices, we study both the coalescence of an initially spread field(interpreted as roughening), and the statistical steady-state properties. Wedistinguish two main size-dependent regimes, depending on the strength of thediffusion term and on the lattice size. Using numerical simulations andmean-field approach, we study the statistics of the field. For weak diffusion,we unveil a characteristic hierarchical structure of the field. We also connectthe model and the iterated function systems concept.

Lattice models of random advection and diffusion and their statistics