A Markov Process on Cyclic Words

University dissertation from Stockholm : KTH Royal Institute of Technology

Author: Erik Aas; Kth.; [2014]


Abstract: The TASEP (totally asymmetric simple exclusion process) studied here is a Markov chain on cyclic words over the alphabet{1,2,...,n} given by at each time step sorting an adjacent pair of letters chosen uniformly at random. For example, from the word 3124 one may go to 1324, 3124, 3124, 4123 by sorting the pair 31, 12, 24, or 43.Two words have the sametype if they are permutations of each other. If we restrict TASEP to words of some particular type m we get an ergodic Markov chain whose stationary distribution we denote by ?m. So?m (u) is the asymptotic proportion of time spent in the state u if the chain started in some word of type m. The distribution ? is the main object of study in this thesis. This distribution turns out to have several remarkable properties, and alternative characterizations. It has previously been studied both from physical, combinatorial, and probabilitistic viewpoints.In the first chapter we give an extended summary of known results and results in this thesis concerning ?. The new results are described (and proved) in detail in Papers I - IV.The new results in Papers I and II include an explicit formula for the value of?at sorted words and a product formula for decomposable words. We also compute some correlation functions for ?. In Paper III we study of a generalization of TASEP to Weyl groups. In Paper IV we study a certain scaling limit of ?, finding several interesting patterns of which we prove some. We also study an inhomogenous version of TASEP, in which different particles get sorted at different rates, which generalizes the homogenous version in several aspects. In the first chapter we compute some correlation functions for ?