Dissertation

Random Walks in Changing Environments

Lelli, Andrea

2021

Overview

This thesis deals with Random Walks on graphs that change over time in a random manner, more precisely we analyse Random Walks on Dynamical Percolation. In this model, the edges of a graph G are either open or closed and refresh their status at rate μ independently from all other edges, while at the same time a random walker moves on G at rate 1 but only along edges which are open. In Chapter 3 we present the known results about mixing time for random walks in dynamical percolation and we give a sketch of the proof of the upper bound for the mixing time of Random Walk on Dynamical Percolation when Gn = T_d^n for all p < p_c discussed in detail in [PSS15]. Later, we show a bound on the mixing time of the Random-Cluster model for lattices with polynomial growth. Finally, we introduce Random Walks on Dynamical Random Cluster. This model is similar to the Random Walks on Dynamical Percolation with the only difference that the refresh of the edges depends on the configuration of the open edges in the graph at the time of the update. We prove that on the d-dimensional torus with side length n, in the subcritical regime, the mixing time for the full system is bounded above by n^2/μ up to constants.

• Publisher: ProQuest Dissertations & Theses
• Subject: Graphs