Unimodularity in randomly generated graphs / Florian Sobieczky, editor.

Format:

Book

Contributor:

Sobieczky, Florian, editor.

Series:

Contemporary mathematics (American Mathematical Society). 0271-4132 719

Contemporary mathematics, 719 0271-4132

Language:

English

Subjects (All):

Random graphs.

Graph theory.

Physical Description:

1 online resource (226 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2018]

Summary:

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.

Contents:

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

2. Basic constructions

3. A discontinuous phase transition

4. Nonamenability and uniqueness

Invariant -percolation on regular trees

Sparse graph limits along balls

1. Hyperfiniteness

2. Yu's Property

3. Further questions that arise

4. An infinite model

Percolation and coarse conformal uniformization

2. Two conjectures

3. Conformal invariance and hyperbolicity

Invariant tilings and unimodular decorations of Cayley graphs

2. Duality

Distributional lattices on Riemannian symmetric spaces

2. Distributional lattices

3. General Properties of Poisson-Voronoi tilings in Symmetric Spaces

4. Additional structure for Poisson-Voronoi tessellations in nonpositively curved spaces

Eternal Family Trees and dynamics on unimodular random graphs

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

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

All properly ergodic Markov chains over a free group are orbit equivalent

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

Shift-coupling of random rooted graphs and networks

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

Back Cover.

Notes:

"AMS Special Session on Unimodularity in Randomly Generated Graphs, October 8-9, 2016, Denver, Colorado."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-5017-8