Metastability is a phenomenon where a dynamical system can move between different states that are not its global equilibrium state. On short time scales the system can find itself equilibrized in a...Show moreMetastability is a phenomenon where a dynamical system can move between different states that are not its global equilibrium state. On short time scales the system can find itself equilibrized in a certain region of its state space (a local equilibrium), whereas on a long time scale it will make quick transitions between new, different regions of its state space. These local equilibria are referred to as the metastable states. One of the uses of metastability is for model reduction. In this thesis we will restrict ourselves to Markovian processes and consider the networks associated to the transitions of the Markov chains. Instead of considering a Markov process on a very large state space, one can look at the process on a reduced state space representing these metastable states. The idea is that this coarse-grained network "mimics" the behaviour of the original network. We shall give two different mathematical definitions for metastability of Markov chains. In most cases where metastability is studied, limiting asymptotics are wielded. One must think of taking limits of large volume or low temperature. However in the paper [1] by Avena, Castell, Gaudillière and Mélot a new framework is introduced by which to describe "metastability" without the use of these limits. The network of transitions of a given Markov process is coarse-grained to a state space that represents probability measures which focus on different regions of the original finite state space (the local equilibria). It does so through the use of intertwining dualities. We say that a n × n-matrix A is intertwined with a m × m-matrix C with respect to a m × n-matrix B if BA = CB. For our discussion we are given a Markov process on a finite state space with an associated transition matrix P in order to find another Markov process on a smaller state space with transition matrix P and a matrix Λ such that ΛP = PΛ where the rows of Λ are probability measures on the original state space (representing the local equilibria). In this thesis we will explore this framework based on intertwining on a toy model consisting of three nodes that we want to reduce to a network of two nodes. The goal is to illustrate the method in [1] in this explicit model and explore which evolutions among the local equilibria can be described; how this relates to the spectrum of the transition matrix P of the Markov chain in this model; and its implications on the mixing time.Show less
In this thesis we will discuss the connection between Markov chains and electrical networks. We will explain how weighted graphs are linked with Markov chains. We will then present classical...Show moreIn this thesis we will discuss the connection between Markov chains and electrical networks. We will explain how weighted graphs are linked with Markov chains. We will then present classical results regarding the connection between reversible Markov chains and electrical networks. Based on recent work, we can also make a connection between general Markov chains and electrical networks, where we also show how the associated electrical network can be physically interpreted, which is based on a nonexisting electrical component. We will then especially see which specific results still apply for these general Markov chains. This thesis concludes with an application based on this connection, called the Transfer current theorem. Proving this theorem relies upon the connection between random spanning trees with Markov chains and electrical networks.Show less
