Gibbsian properties: examples of failures and an application

PhD ceremony: Mr. V. Ermolaev, 16.15 uur, Aula Academiegebouw, Broerstraat 5, Groningen

Dissertation: Gibbsian properties: examples of failures and an application

Promotor(s): prof. C. Külske, prof. A.C.D. van Enter en prof. E. Verbitsky

Faculty: Mathematics and Natural Sciences

The present thesis investigates further under which conditions the transformation of the originally Gibbs system preserves or loses its Gibbs property. This question is addressed for systems living on a complete graph or a tree. The former type of system is referred to as a mean-field system. In either case we consider Bernoulli random variables (spins) at nodes of the graph taking either +1 or -1 values. The transformations of interest are simple spin-flip dynamics altering value of each of the spins to an opposite one. For tree models we discuss different mechanisms leading to non-Gibbsianness revealing previously not discovered type of non-Gibbsianness, when any configuration within a finite region is a discontinuity point. This result underlines the importance of the size of a boundary of a finite region, this is to be contrasted with similar models living on lattices. Further we draw attention to a mean-field model in vanishing external magnetic field. In this case there are two governing parameters: the initial temperature of the system and the temperature of the dynamics. The relative simplicity of the problem allows to decompose the ‘time x initial temperature’ plane at fixed dynamical temperature into regions where conditional distributions have discontinuities and where they do not.  

Finally, the last part standing relatively apart from the main topic discusses connection between variable length Markov chains and Gibbs measures. We apply the Gibbs formalism for classification problems (where VLMC were of great use) and see the superiority of the Gibbsian approach.