Random walks, random configurations, and horocyclic products Proposal for an FWF project by Wolfgang WOESS Institut f¨ ur Mathematik C (Mathematical Structure Theory) Technische Universit¨at Graz December 30, 2005 Contents 1. Introduction 1 2. Recent and current work, projects, and collaborators 2 3. State of the art 5 4. Project research: details 7 5. Project duration 10 6. Project personnel 10 7. International collaboration 11 8. Research plan, dissemination 13 9. Importance and impact 13 10. Requested funding 14 11. References 15 12. Curriculum vitae and recent publications of Wolfgang Woess 18 13. Curriculum vitae and publications of James Parkinson 21 14. Curriculum vitae and publications of Wilfried Huss 23 1. Introduction After returning to Austria in fall, 1999 (from 1988 to 1999, I had been professor at the University of Milan, Italy), I started to build up a small research group in my area. In view of the fact that the (few) assistant positions at Institut f¨ ur Mathematik C were already held by tenured persons working in different fields, this could an can only be accomplished via project work, by engagement of PostDocs and PhD students, in combination with efforts to bring established workers in my field to Graz for shorter periods as visiting professors, or within other exchange programs (Socrates/Erasmus, RDSES-program of ESF,...). Several young people have been working in my projects here in Graz, see below in §2. However, only one regular assistant position became vacant in the last 6 years, so that the continuation of research activities within future projects is vital for the group. At the center of my efforts was the previous project “Asymptotic properties of random walks on graphs”, FWF P15577, of which the one proposed here is to be considered a direct continua- tion along more specific lines. (One of my current project collaborators - Dr. Florian Sobieczky - has also presented his own new proposal which officially, according to FWF rules, figures as a successor proposal to FWF P15577, but goes in a rather different direction than the present one.) The original proposal text of FWF P15577 is still available online at the web address http://www.math.tugraz.at/∼woess/fwf.ps so that here I shall avoid a long outline of the general state of art of random walk theory. The basic philosophy is to study the interplay between stochastic properties of random walks on graphs and groups (which are typically infinite and discrete in my case) and analytic aspects 1