Introduction to (generalized) Gibbs measures

TL;DR

This paper provides a rigorous mathematical introduction to Gibbs and generalized Gibbs measures on lattices, emphasizing the importance of continuity properties of conditional probabilities for a proper theoretical framework.

Contribution

It offers a detailed, mathematically rigorous presentation of Gibbs measures, highlighting the significance of continuity properties over traditional potential-based approaches.

Findings

Emphasizes the role of continuity of conditional probabilities in Gibbs measures.

Clarifies the distinction between Gibbs and generalized Gibbs measures.

Provides a comprehensive mathematical foundation for equilibrium states in lattice systems.

Abstract

These notes have been written to complete a mini-course "Introduction to (generalized) Gibbs measures" given at the universities UFMG (Universidade Federal de Minas Gerais, Belo Horizonte, Brasil) and UFRGS (Universidade Federal do Rio Grande do Sul, Porto Alegre, Brasil) during the first semester 2007. The main goal of the lectures was to describe Gibbs and generalized Gibbs measures on lattices at a rigorous mathematical level, as equilibirum states of systems of a huge number of particles in interaction. In particular, our main message is that although the historical approach based on potentials has been rather successful from a physical point of view, one has to insist on (almost sure) continuity properties of conditional probabilities to get a proper mathematical framework.