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These lecture notes study the interplay between randomness and geometry of graphs. The first part of the notes reviews several basic geometric concepts, before moving on to examine the manifestation of the underlying geometry in the behavior of random processes, mostly percolation and random walk. The study of the geometry of infinite vertex transitive graphs, and of Cayley graphs in particular, is fairly well developed. One goal of these notes is to point to some random metric spaces modeled by graphs that turn out to be somewhat exotic, that is, they admit a combination of properties not encountered in the vertex transitive world. These include percolation clusters on vertex transitive graphs, critical clusters, local and scaling limits of graphs, long range percolation, CCCP graphs obtained by contracting percolation clusters on graphs, and stationary random graphs, including the uniform infinite planar triangulation (UIPT) and the stochastic hyperbolic planar quadrangulation (SHIQ).

Mathematics --- Physical Sciences & Mathematics --- Geometry --- Mathematics. --- Geometry. --- Probabilities. --- Graph theory. --- Physics. --- Statistics. --- Continuum mechanics. --- Probability Theory and Stochastic Processes. --- Mathematical Methods in Physics. --- Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences. --- Continuum Mechanics and Mechanics of Materials. --- Graph Theory. --- Stochastic processes. --- Geometric probabilities. --- Probabilities --- Random processes --- Distribution (Probability theory. --- Mathematical physics. --- Mechanics. --- Mechanics, Applied. --- Solid Mechanics. --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Physical mathematics --- Distribution functions --- Frequency distribution --- Characteristic functions --- Euclid's Elements --- Statistics . --- Graph theory --- Graphs, Theory of --- Theory of graphs --- Combinatorial analysis --- Topology --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Extremal problems

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Statistical science --- statistisch onderzoek

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This volume is dedicated to the memory of the late Oded Schramm (1961-2008), distinguished mathematician. Throughout his career, Schramm made profound and beautiful contributions to mathematics that will have a lasting influence. In these two volumes, Editors Itai Benjamini and Olle Häggström have collected some of his papers, supplemented with three survey papers by Steffen Rohde, Häggström and Cristophe Garban that further elucidate his work. The papers within are a representative collection that shows the breadth, depth, enthusiasm and clarity of his work, with sections on Geometry, Noise Sensitivity, Random Walks and Graph Limits, Percolation, and finally Schramm-Loewner Evolution. An introduction by the Editors and a comprehensive bibliography of Schramm's publications complete the volume. The book will be of especial interest to researchers in probability and geometry, and in the history of these subjects.

Statistical science --- statistisch onderzoek