Markov random fields and applications (series of lectures - part II)
A talk in the Cluster Group Stochastic Analysis series by
Tetyana Pasurek
| Abstract: | Lecture II: Existence and uniqueness problems for Markov fields: Comparing Dobrushin's and Ruelle's approaches.
Markov fields appear as Gibbs equilibrium states in statistical mechanics or as Markov networks or undirected graphical models in Big Data analysis and machine learning. The goal is to develop a unified theory that can then be applied to specific models (classical or quantum; on lattices Zd and general graphs or in the continuum Rd), covering a large amount of the results known so far. In particular, we address the problems of existence and uniqueness of Markov fields, their mixing properties and dimension-free estimates of convergence rates. We also connect the two basic approaches - Dobrushin's theory of weak dependence and Ruelle's superstability estimates - and extend them to unbounded interactions and irregular underlying spaces.
We discuss the existence and uniqueness problems for Markov fields in the framework of Dobrushin's contraction method and its local versions. As a by-product of our approach (or as a particularly simple case), we provide an elementary and self-contained proof of Harris' ergodic theorem for Markov chains in the spirit of Hairer and Mattingly (2008). |