Dynamic Phenomena in Interacting Particle Systems: Phase Transitions and Equilibrium

TL;DR

This thesis explores phase transitions and equilibrium states in stochastic particle systems, introducing new methods and results for activated random walks, contact processes, and quasi-stationary distributions.

Contribution

It provides novel proofs of slow phases, establishes invariant measures at critical parameters, and proves existence and uniqueness of quasi-stationary distributions in complex models.

Findings

Existence of slow phase for large sleep rates in activated random walks.

Invariant measure exists for modified contact process at critical infection rates.

Unique quasi-stationary distributions are proven for certain subcritical processes.

Abstract

This thesis investigates critical phenomena and equilibrium states in various stochastic models through three interconnected studies. In the first chapter, we analyze the Activated Random Walk model on a one-dimensional ring in the high-density regime. We introduce a toppling procedure that incrementally constructs an environment demonstrating the sustained activity over extended periods. This approach provides a concise and self-contained proof of the existence of a slow phase for arbitrarily large sleep rates. The second chapter focuses on a modified unidimensional contact process with varying infection rates. Specifically, infection spreads at rate $\lambda_e$ at the boundaries of the infected region and at rate $\lambda_i$ elsewhere. We establish the existence of an invariant measure for this process when $\lambda_i=\lambda_c$, $\lambda_e=\lambda_c+\varepsilon$ where $\lambda_c$ denotes the critical parameter for the standard contact process. Furthermore, we demonstrate that the process, when observed from the right edge, converges weakly to this invariant measure. We also show that infection dies almost surely along the critical curve within the attractive region of the phase space. In the final chapter, we explore quasi-stationary distributions (QSDs) for two subcritical population processes in continuous time: branching random walks and branching processes with genealogy. We prove the existence and uniqueness of QSDs for these processes by leveraging spatial aspects of their dynamics.