Hydrodynamic limit and fluctuations of the zero range process on a randomly oriented graph

Abstract

This thesis is concerned with interacting particle systems on asymmetric random graphs. Motivated by results on non-reversible random walks in random environments obtained by Kozma and Toth, our aim is to study interacting particle systems on graphs subject to comparable conditions. We study the mean-zero zero range process on a randomly oriented graph with vertex set Z × {-1,1} designed to be one of the simplest graphs for which a nontrivial bistochastic random orientation is possible. Using the relative entropy method, we prove that the hydrodynamic limit of the process is the heat equation. In doing so we introduce a local equilibrium measure that decomposes the graph into components of constant density in order to eliminate non-gradient terms.

We then prove that the order of the current fluctuations is bounded above by t^{1/2} and obtain a lower bound for the displacement of the second class particle. The upper bound is only valid in a quenched random environment, whereas the lower bound is only valid in an annealed random environment. It does not appear possible to combine these two results to prove the exact order of the fluctuations in either type of random environment.

Date of Award	25 Jan 2022
Original language	English
Awarding Institution	University of Bristol
Supervisor	Marton Balazs (Supervisor) & Balint A Toth (Supervisor)