Random geometry and vertex-reinforced jump processes
This thesis comprises three papers studying several mathematical models related to geometric Markov processes and random processes with reinforcements. The main goal of these works is to investigate the dynamics as well as the limiting behaviour of the models as time goes to infinity, the existence of invariant measures and limiting distributions, the speed of convergence and other interesting relevant properties.
We firstly discuss in the chapter Introduction the background: products of random matrices, asymptotic pseudo-trajectories and Markov chains in a general state space. We then outline motivation and overview of the main results in the papers included in this thesis.
In the first paper, we deal with a Markov chain model of convex polygons, which are random consecutive subdivisions of an initial convex polygon. Applying the theory of products of random matrices, we prove the universal convergence of these random convex polygons to a ``flat figure". Beside this, we present a discussion about the speed of convergence and the computation of invariant measure in the case of
In the second paper, we investigate a model of strongly vertex-reinforced jump processes (VRJP). Using the method of stochastic approximation, we show the connection between strongly VRJP and an asymptotic pseudo-trajectory of a vector field in order to study the dynamics of the model. In particular, we prove that the strongly VRJP on a complete graph will almost surely have an infinite local time at one vertex, while the local times at all the remaining vertices remain bounded.
In the last paper, we consider a class of random walks taking values in simplexes and study the existence of limiting distributions. In some special cases of Markov chain models, we prove that the limiting distributions are Dirichlet. In addition, we introduce a related history-dependent random walk model in [0,1] based on Friedman’s urn-type schemes and show that this random walk converges in distribution to the arcsine law.