Beyond the Gibbs Paradigm Chapman & Hall/CRC Monographs and Research Notes in Mathematics Series

Gibbs (or DLR) measures are the main objects in classical equilibrium statistical mechanics. Statistical mechanics deals with models from mathematical physics and chemistry where one is interested in, for example, some average behaviour of an interacting system subjected to some noise. They were originally introduced as probability measures on systems of infinitely many particles in infinite volume, satisfying a set of consistent conditional probabilities. Probability measures captured the uncertainty or noise of the state of the system. Gibbs measures also play a role in various other domains, such as Dynamical Systems, ergodic theory, spatial statistics and pattern recognition.

Motivation: why going beyond Gibbs measures. History. Reviewing the Gibbs measure formalism. Transformations of Gibbs measures. Decimation of Gibbs measures. Projections. Time-evolution of Gibbs measures. Local coarse-graining. A quantum excursion. Disorder and Gibbs-non-Gibbs problems. Spin models. Random walks in random environments. Further examples. Random-cluster models. Applications: Denoising algorithms and Gibbs measures. Dobrushin’s restoration program. Weakly Gibbs measures. Almost Gibbs measures, intuitively weak Gibbs measures. Conclusions.