These proceedings represent the current state of research on the topics 'boundary theory' and 'spectral and probability theory' of random walks on infinite graphs. They are the result of the two workshops held in Styria (Graz and St. Kathrein am Offenegg, Austria) between June 29th and July 5th, 2009. Many of the participants joined both meetings. Even though the perspectives range from very different fields of mathematics, they all contribute with important results to the same wonderful topic from structure theory, which, by extending a quotation of Laurent Saloff-Coste, could be described...

Seit uber 20 Jahren gibt es im Suden der chinesischen Provinz Sichuan keine Bienen mehr, weil der naturliche Kreislauf a " wahrscheinlich durch den Einsatz von Pestiziden a " zerstA

rt wurde. Zahlreiche Blumenarten sind verschwunden und die Bauern mussen seitdem die Bluten der Birnbaume muhsam von Hand bestauben. Vor diesem Hintergrund wird die Geschichte der kleinen Lian erzahlt, die im a Tal der Birnena auf dem Hof ihrer GroA eltern lebt. Erst an ihrem zehnten Geburtstag erfahrt Lian, dass es uberhaupt Bienen gibt, und sie ist sofort fasziniert von den kleinen, nutzlichen Arbeiterinnen....

What is order which is not based on simple repetition, that is, periodicity? How must atoms be arranged in a material so that it diffracts like a quasicrystal? How can we describe aperiodically ordered systems mathematically? Originally triggered by the later nobel prize winning discovery of quasicrystals, the investigation of aperiodic order has by now become a well-established and strongly evolving field of mathematical research. It is closely tied to a surprising variety of branches of mathematics and physics. The book offers an overview over the state of the art in the field of aperiodic...