This is a volume of lecture notes based on three series of lectures given by three visiting professors of RIMS, Kyoto University during the year long project research "Discrete Geometric Analysis" in the Japanese academic year 2012. The aim of the project research was to make comprehensive research on topics related to discreteness in geometry, analysis and optimization. The goal is to expand and make a new stream of discrete geometric analysis by exchanging ideas in each area. The main themes during the project and also of this volume are threefold: i) Discrete probability theory and analysis on graphs, ii) Discrete convex analysis and its applications, iii) Geometric group theory. We had five conferences at Kyoto University in the project research. The three visiting professors gave the series of lectures on the following topics:

• Loop Erased Walks and Uniform Spanning Trees, by Martin T. Barlow (University of British Columbia)

• Combinatorial Rigidity: Graphs and Matroids in the Theory of Rigid Frameworks, by Tibor Jordán (Eötvös University)

• Analysis and Geometry on Groups, by Andrzej Zuk (Université Paris 7)