Metastability for a PDE with blow-up and the FFG dynamics on diluted models

TL;DR

This thesis investigates the stability of probability models under small perturbations, demonstrating instability in dynamical systems and stability of Gibbs measures in statistical mechanics, with implications for understanding phase transitions.

Contribution

It provides a comparative analysis of stability in dynamical systems and Gibbs measures, highlighting conditions for stability and instability under small perturbations.

Findings

Random perturbations destabilize deterministic systems.

Gibbs measures are stable under small parameter changes.

Certain configurations remain stable at zero temperature.

Abstract

This thesis consists of two separate parts: in each we study the stability under small perturbations of certain probability models in different contexts. In the first, we study small random perturbations of a deterministic dynamical system and show that these are unstable, in the sense that the perturbed systems have a different qualitative behavior than that of the original system. In the second part we situate ourselves in the context of statistical mechanics, where we study the stability of equilibrium infinite-volume measures under small deterministic perturbations in the parameters of the model. More precisely, we show that Gibbs measures for a general class of systems are continuous with respect to changes in the interaction and/or density of particles and, hence, stable under small perturbations of them. We also study under which conditions do certain typical configurations of these systems remain stable in the zero-temperature limit.