Course Notes for EE 87021 Advanced Topics in Random Wireless Networks Martin Haenggi November 1, 2010 c Martin Haenggi, 2010 ii Contents I Point Process Theory 1 1 Introduction 3 1.1 Motivation . 3 1.2 Asymptotic Notation . 3 2 Description of Point Processes 5 2.1 One-dimensional Point Processes . 5 2.2 General Point Processes . 5 2.3 Basic Point Processes . 6 2.3.1 One-dimensional Poisson processes . 6 2.3.2 Spatial Poisson processes . 7 2.3.3 General Poisson point processes . 7 2.4 Distributional Characterization . 7 2.4.1 The distribution of a point process . 8 2.4.2 Comparison with numerical random variables . 8 2.4.3 Distribution of a point process viewed as a random set . 9 2.4.4 Finite-dimensional distributions and capacity functional . 9 2.4.5 Measurable decomposition . 10 2.4.6 Intensity measure . 11 2.5 Properties of Point Processes . 11 2.6 Point Process Transformations . 12 2.6.1 Shaping an inhomogeneous Poisson point process . 12 2.7 Distances . 14 2.8 Marked point processes . 14 3 Sums over Point Processes 17 3.1 Notation . 17 3.2 Campbell’s Theorem for the Mean . 18 3.3 The Probability Generating Functional . 19 3.3.1 The moment-generating function of the sum . 19 3.3.2 The characteristic or Laplace functional for the Poisson point process . 21 3.3.3 The probability generating functional for the Poisson point process . 21 3.3.4 Relationship between moment-generating function and the functionals . 22 3.4 Applications . 23 3.4.1 Mean interference in stationary point process . 23 iii 3.4.2 Variance of the interference in stationary Poisson point process . 25 3.4.3 Interference from nearest transmitters . 25 3.4.4 Interference distribution without fading . 26 3.4.5 Interference distribution with fading . 28 3.4.6 Outage in Poisson networks with Rayleigh fading . 29 3.5 Stable Distributions . 30 3.5.1 Definition . 30 3.5.2 LePage Series representation . 30 3.5.3 Shot noise . 30 4 Moment Measures of Point Processes 31 4.1 Introduction . 31 4.2 The First-Order Moment Measure . 31 4.2.1 Constant density vs. stationarity . 32 4.3 Second Moment Measures . 32 4.4 Second Moment Density . 35 4.5 Second Moments for Stationary Processes . 39 5 Conditioning and Palm Theory 43 5.1 Introduction and Basic Concepts for Stationary Point Processes . 43 5.1.1 The local approach . 43 5.1.2 The global approach . 45 5.2 The Palm Distribution . 45 5.2.1 Heuristic introduction . 45 5.2.2 A first definition of the Palm distribution (stationary point processes) . 46 5.2.3 A second definition of the Palm distribution (general point processes) . 47 5.2.4 Alternative interpretation and conditional intensity . 50 5.3 The Reduced Palm Distribution . 50 5.3.1 Definition . 50 5.3.2 Palm distribution for PPPs and proof of Slivnyak’s theorem . 51 5.3.3 Isotropy of Palm distributions . 52 5.3.4 Palm expectations . 52 5.4 Second Moments Measures and Palm Distributions for Stationary Processes . 52 II Percolation Theory, Connectivity, and Coverage 55 6 Introduction 57 6.1 Motivation . 57 6.2 What is Percolation? . 57 7 Bond and Site Percolation 59 7.1 Random Trees and Branching Processes . 59 7.1.1 Percolation on regular branching trees . 59 7.1.2 Generalization to branching processes . 59 7.1.3 Site percolation on the branching tree . 64 7.1.4 Mean cluster size . 64 iv 7.2 Preliminaries for Bond Percolation on the Lattice . 65 7.3 General Behavior of the Percolation Probability . 66 7.3.1 The d-dimensional case . 66 7.3.2 Simple bounds on the percolation probability for the two-dimensional case . 67 7.3.3 Generalization of the lower bound . 69 7.4 Basic Techniques . 70 7.4.1 The FKG Inequality . 70 7.4.2 The BK Inequality . 72 7.4.3 Russo’s Formula . 73 7.4.4 The square-root trick . 75 7.5 Critical Threshold for Bond Percolation on the Square Lattice . 75 7.5.1 Subcritical Phase: Exponential decrease of the radius of the mean cluster size . 75 7.5.2 Supercritical phase: Uniqueness of the infinite open cluster . 76 7.5.3 Critical Threshold . 77 7.5.4 Ingredients used . 79 7.5.5 The Missing Theorem . 79 7.6 Further Results . 79 7.6.1 At the Critical Point . 79 7.6.2 Generalization to d dimensions. 80 7.7 Site Percolation . 81 7.7.1 Results . 82 7.7.2 Numerical Bounds . 83 8 Continuum Percolation 85 8.1 Gilbert’s Disk Graph . 85 8.1.1 Bounding using bond percolation on the square lattice . 85 8.1.2 Bounding using site percolation on the triangular lattice . 85 8.1.3 A better lower bound . 88 8.2 Other Percolation Models . 89 8.2.1 Interference Graph . 89 8.2.2 Secrecy Graph . 90 9 Connectivity 91 9.1 Full Connectivity . 91 9.2 More General Continuum Models . 92 9.2.1 Connectivity: Random connection model (RCM) . 92 9.3 (Abstract) Random Graphs . 93 10 Coverage 95 10.1 Germ-grain (Boolean) models . 95 10.1.1 Uniqueness of the infinite component . 95 v vi Part I Point Process Theory 1 Chapter 1 Introduction In the first part of the course, we give an introduction to stochastic geometry and, in particular, point process theory. 1.1 Motivation Basic questions: How to describe a (random) collection of points in two or three dimensions? • How about in one dimension? (In one dimension, renewal theory helps deal with processes with • independent increments. How to generalize from one dimension to two dimensions? What is the main difference? • Other random geometric objects: How to.