Geometric Gibbs theory

An Introduction to Geometric Gibbs Theory

This is an article I wrote for Dynamics, Games, and Science. In Dynamics, Game, and Science, one of the most important equilibrium states is a Gibbs state. The deformation of a Gibbs state becomes an important subject in these areas. An appropriate metric on the space of underlying dynamical systems is going to be very helpful in the study of deformation. The Teichmüller metric becomes a natura...

Geometric Ergodicity of Gibbs Samplers

Due to a demand for reliable methods for exploring intractable probability distributions, the popularity of Markov chain Monte Carlo (MCMC) techniques continues to grow. In any MCMC analysis, the convergence rate of the associated Markov chain is of practical and theoretical importance. A geometrically ergodic chain converges to its target distribution at a geometric rate. In this dissertation,...

Gibbs Delaunay tessellations with geometric hardcore conditions

Function Models for Teichmüller Spaces and Dual Geometric Gibbs Type Measure Theory for Circle Dynamics

Geometric models and Teichmüller structures have been introduced for the space of smooth circle endomorphisms and for the space of uniformly symmetric circle endomorphisms. The latter one is the completion of the previous one under the Techmüller metric. Moreover, the spaces of geometric models as well as the Teichmüller spaces can be described as the space of Hölder continuous scaling function...

Teichmüller Structures and Dual Geometric Gibbs Type Measure Theory for Continuous Potentials

The Gibbs measure theory for smooth potentials is an old and beautiful subject and has many important applications in modern dynamical systems. For continuous potentials, it is impossible to have such a theory in general. However, we develop a dual geometric Gibbs type measure theory for certain continuous potentials in this paper following some ideas and techniques from Teichmüller theory for ...