Random walks and Markov chains

Random Walks on Finite Quantum Groups

1 Markov chains and random walks in classical probability . . 3 2 Quantum Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Random walks on comodule algebras . . . . . . . . . . . . . . . . . . . . . . 7 4 Random walks on finite quantum groups . . . . . . . . . . . . . . . . . . 11 5 Spatial Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . ....

Random Walks on Infinite Graphs and Groups — a Survey on Selected Topics

Contents 1. Introduction 2 2. Basic definitions and preliminaries 3 A. Adaptedness to the graph structure 4 B. Reversible Markov chains 4 C. Random walks on groups 5 D. Group-invariant random walks on graphs 6 E. Harmonic and superharmonic functions 6 3. Spectral radius, amenability and law of large numbers 6 A. Spectral radius, isoperimetric inequalities and growth 6 B. Law of large numbers 9 ...

Semigroups, Rings, and Markov Chains

We analyze random walks on a class of semigroups called ``left-regular bands.'' These walks include the hyperplane chamber walks of Bidigare, Hanlon, and Rockmore. Using methods of ring theory, we show that the transition matrices are diagonalizable and we calculate the eigenvalues and multiplicities. The methods lead to explicit formulas for the projections onto the eigenspaces. As examples of...

Random walks on directed Cayley graphs

Previous authors have shown bounds on mixing time of random walks on finite undirected Cayley graphs, both with and without self-loops. We extend this to the most general case by showing that a similar bound holds for walks on all finite directed Cayley graphs. These are shown by use of two new canonical path theorems for mixing time of non-reversible Markov chains. The first result is related ...

Reversible Markov Chains and Random Walks on Graphs

Bounds on Biased and Unbiased Random Walks

We analyze several random random walks on one-dimensional lattices using spectral analysis and probabilistic methods. Through our analysis, we develop insight into the pre-asymptotic convergence of Markov chains.