Lecture Notes on Random Geometric Models --- Random Graphs, Point Processes and Stochastic Geometry

DoctoralThe goal of these lessons is to provide a quick access to some popular models of random geometric structures arising in a number of used in applications: communication networks (including social, transportation and wireless networks), geology, material sciences and astronomy. The course is composed of the following 15 lectures:1. Bond percolation on the square lattice2. Galton-Watson tree3. Erdős-Rényi graph — emergence of the giant component4. Graphs with a given node degree distribution5. Typical nodes and random unimodular graphs6. Erdős-Rényi graph — emergence of the full connectivity7. Poisson point process8. Point conditioning and Palm theory for point processes9. Hard-core point processes10. Stationary point processes and mass transport principle11. Stationary Voronoi tessellation12. Ergodicity and point-shift invariance13. Random closed sets14. Boolean model and coverage processes15. Connectedness of random sets and continuum percolationUsually, these topics are treated in distinct monographs: random graphs (lessons 2–6), point processes (7-12), stochastic geometry (13-14), where the percolation models presented in lesson 1 and 15 are often addressed separately. Gathering them in a single course provides an opportunity to observe some similarities and even fundamental relations between different models