Coin-Turning, Random Walks and Inhomogeneous Markov Chains

This research monograph explores new frontiers in Markov chains. Although time-homogeneous Markov chains are well understood, this is not at all the case with time-inhomogeneous ones. The book, after a review on the classical theory of homogeneous chains, including the electrical network approach, introduces several new models which involve inhomogeneous chains as well as related new types of random walks (for example, "coin turning", "conservative" and "Rademacher" walk). Scaling limits, the breakdown of the classical limit theorems as well as recurrence and transience are investigated. The relationship with urn models is the subject of two chapters, providing additional connections to other parts of probability theory. Random walks on random graphs are discussed as well, as an area where the method of electric networks is especially useful. This is illustrated by presenting random walks in random environments and random labyrinths. The monograph puts emphasis on showing examples and open problems besides providing rigorous analysis of the models. Several figures illustrate the main ideas, and a large number of exercises challenge the interested reader.

János Engländer received his Diploma in Mathematics at Eötvös Loránd University, Budapest, Hungary and his PhD in Mathematics from the Technion, Haifa, Israel. He worked as postdoctoral fellow at the Weierstrass Institute (Berlin) and at EURANDOM (Eindhoven) before joining the University of California and later the University of Colorado, where he is currently a professor at the Department of Mathematics. He is the author of a research monograph on spatial branching processes, and his research interests include discrete and continuous probability theory and partial differential equations.

Stanislav Volkov received his Diploma in Mathematics and PhD in Mathematics from Moscow State University, Russia, and his MA and MSc in Economics from New Economics School, Russia and University of Wisconsin–Madison, USA. He worked as postdoctoral fellow in the Fields Institute (Canada) and EURANDOM (Eindhoven) before moving to a permanent position in the University of Bristol (UK). Since 2012, he is a Professor of Mathematical Statistics in Lund University, Sweden. His research interests include discrete probability, history-dependent random walks, and percolation.