Hydrodynamic Scaling, Convex Duality and Asymptotic Shapes of Growth Models
1998, v.4, Issue 1, 1-26
We present a technique for simultaneously deriving two related results: hydrodynamic scaling limits for one-dimensional totally asymmetric particle systems and asymptotic shapes for growth models. The idea is to specify the particle dynamics in terms of a microscopic Lax - Oleinik formula which leads directly to the macroscopic description in terms of a nonlinear conservation law. The law of large numbers required for this link comes from the growth model that is embedded in the particle system. In the limit, the asymptotic shape of the growth model becomes the convex conjugate of the flux of the conservation law, and the latter is computable from the particle system in equilibrium. The asymptotic shape is then obtained from the duality relation. The method is illustrated with three examples.
Keywords: hydrodynamic scaling limit,growth model,Lax - Oleinik formula,convex duality