Lattice models of random advection and diffusion and their statistics

We study in detail a one-dimensional lattice model of a continuum, conserved field (mass) that is transferred deterministically between neighboring random sites. The model belongs to a wider class of lattice models capturing the joint effect of random advection and diffusion and encompassing as specific cases some models studied in the literature, such as those of Kang-Redner, Kipnis-Marchioro-Presutti, Takayasu-Taguchi, etc. The motivation for our setup comes from a straightforward interpretation of the advection of particles in one-dimensional turbulence, but it is also related to a problem of synchronization of dynamical systems driven by common noise. For finite lattices, we study both the coalescence of an initially spread field (interpreted as roughening), and the statistical steady-state properties. We distinguish two main size-dependent regimes, depending on the strength of the diffusion term and on the lattice size. Using numerical simulations and a mean-field approach, we study the statistics of the field. For weak diffusion, we unveil a characteristic hierarchical structure of the field. We also connect the model and the iterated function systems concept.