Unimodularity in Randomly Generated Graphs.

Cover -- Title page -- Contents -- Preface -- Monotonicity of average return probabilities for random walks in random environments -- 1. Introduction -- 2. Statements of Results and Background -- 3. Proofs -- References -- Counterexamples for percolation on unimodular random graphs -- 1. Introduction -- 2. Basic constructions -- 3. A discontinuous phase transition -- 4. Nonamenability and uniqueness -- References -- Invariant -percolation on regular trees -- References -- Sparse graph limits along balls -- 1. Hyperfiniteness -- 2. Yu's Property -- 3. Further questions that arise -- 4. An infinite model -- References -- Percolation and coarse conformal uniformization -- 1. Introduction -- 2. Two conjectures -- 3. Conformal invariance and hyperbolicity -- References -- Invariant tilings and unimodular decorations of Cayley graphs -- 1. Introduction -- 2. Duality -- References -- Distributional lattices on Riemannian symmetric spaces -- 1. Introduction -- 2. Distributional lattices -- 3. General Properties of Poisson-Voronoi tilings in Symmetric Spaces -- 4. Additional structure for Poisson-Voronoi tessellations in nonpositively curved spaces -- References -- Eternal Family Trees and dynamics on unimodular random graphs -- 1. Introduction -- 2. Unimodular Networks -- 3. Vertex-Shifts and Foil Classification -- 4. Eternal Family Trees -- 5. Trees and Networks Beyond Unimodularity -- 6. Eternal Branching Processes -- Conclusion -- Acknowledgements -- References -- Circular slider graphs: de Bruijn, Kautz, Rauzy, lamplighters and spiders -- Introduction -- 1. De Bruijn graphs -- 2. Circular slider graphs -- 3. Examples -- 4. Periodic slider graphs: connectedness and step induced graphs -- 5. Missing links and transversally Markov circular slider graphs -- 6. Lamplighters over cyclic groups -- 7. Lamplighters and circular slider graphs.

8. Spider slider graphs -- References -- All properly ergodic Markov chains over a free group are orbit equivalent -- 1. Introduction -- 2. Preliminaries -- 3. General results regarding Markov chains -- 4. General constructions of orbit-equivalences -- 5. From properly ergodic to generator-ergodic -- 6. Proof of the main theorem -- References -- Shift-coupling of random rooted graphs and networks -- 1. Introduction -- 2. Random Rooted Graphs and Networks -- 3. Shift-Coupling of Random Rooted Networks -- 4. The Unimodular Case and Balancing Transport Kernels -- 5. Network Extension and Unimodularization -- 6. A Construction Using Stable Transports -- 7. Proofs -- 8. Bibliography of Analogous Results for Point Processes -- Acknowledgements -- References -- Back Cover.

This volume contains the proceedings of the AMS Special Session on Unimodularity in Randomly Generated Graphs, held from October 8-9, 2016, in Denver, Colorado. Unimodularity, a term initially used in locally compact topological groups, is one of the main examples in which the generalization from groups to graphs is successful. The "randomly generated graphs", which include percolation graphs, random Erdős-Rényi graphs, and graphings of equivalence relations, are much easier to describe if they result as random objects in the context of unimodularity, with respect to either a vertex-transient "host"-graph or a probability measure. This volume tries to give an impression of the various fields in which the notion currently finds strong development and application: percolation theory, point processes, ergodic theory, and dynamical systems.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.