Showing papers by "Amir Dembo published in 2016"

Weakly Asymmetric Non-Simple Exclusion Process and the Kardar–Parisi–Zhang Equation

Abstract: We analyze a class of non-simple exclusion processes and the corresponding growth models by generalizing the discrete Cole–Hopf transformation of Gartner (Stoch Process Appl, 27:233–260, 1987). We identify the main non-linearity and eliminate it by imposing a gradient type condition. For hopping range at most 3, using the generalized transformation, we prove the convergence of the exclusion process toward the Kardar–Parisi–Zhang (kpz) equation. This is the first universality result under the weak asymmetry concerning interacting particle systems. While this class of exclusion processes are not explicitly solvable, under the weak asymmetry we obtain the exact one-point limiting distribution for the step initial condition by using the previous result of Amir et al. (Commun Pure Appl Math, 64(4): 466–537, 2011) and our convergence result.

Large Deviations for Diffusions Interacting Through Their Ranks

Abstract: We prove a Large Deviations Principle (LDP) for systems of diffusions (particles) interacting through their ranks, when the number of particles tends to infinity. We show that the limiting particle density is given by the unique solution of the approriate McKean-Vlasov equation and that the corresponding cumulative distribution function evolves according to the porous medium equation with convection. The large deviations rate function is provided in explicit form. This is the first instance of a LDP for interacting diffusions, where the interaction occurs both through the drift and the diffusion coefficients and where the rate function can be given explicitly. In the course of the proof, we obtain new regularity results for a certain tilted version of the porous medium equation.