Thinking Probabilistically: Stochastic Processes, Disordered Systems, and Their Applications

Cover
Half-title
Title page
Copyright information
Contents
Acknowledgments
1 Introduction
	1.1 Probabilistic Surprises
	1.2 Summary
	1.3 Exercises
2 Random Walks
	2.1 Random Walks in 1D
	2.2 Derivation of the Diffusion Equation for Random Walks in Arbitrary Spatial Dimension
	2.3 Markov Processes and Markov Chains
	2.4 Google PageRank: Random Walks on Networks as an Example of a Useful Markov Chain
	2.5 Relation between Markov Chains and the Diffusion Equation
	2.6 Summary
	2.7 Exercises
3 Langevin and Fokker–Planck Equations and Their Applications
	3.1 Application of a Discrete Langevin Equation to a Biological Problem
	3.2 The Black–Scholes Equation: Pricing Options
	3.3 Another Example: The “Well Function” in Hydrology
	3.4 Summary
	3.5 Exercises
4 Escape Over a Barrier
	4.1 Setting Up the Escape-Over-a-Barrier Problem
	4.2 Application to the 1D Escape Problem
	4.3 Deriving Langer\'s Formula for Escape-Over-a-Barrier in Any Spatial Dimension
	4.4 Summary
	4.5 Exercises
5 Noise
	5.1 Telegraph Noise: Power Spectrum Associated with a Two-Level-System
	5.2 From Telegraph Noise to 1/f Noise via the Superposition of Many Two-Level-Systems
	5.3 Power Spectrum of a Signal Generated by a Langevin Equation
	5.4 Parseval’s Theorem: Relating Energy in the Time and Frequency Domain
	5.5 Summary
	5.6 Exercises
6 Generalized Central Limit Theorem and Extreme Value Statistics
	6.1 Probability Distribution of Sums: Introducing the Characteristic Function
	6.2 Approximating the Characteristic Function at Small Frequencies for Distributions with Finite Variance
	6.3 Central Region of CLT: Where the Gaussian Approximation Is Valid
	6.4 Sum of a Large Number of Positive Random Variables: Universal Description in Laplace Space
	6.5 Application to Slow Relaxations: Stretched Exponentials
	6.6 Example of a Stable Distribution: Cauchy Distribution
	6.7 Self-Similarity of Running Sums
	6.8 Generalized CLT via an RG-Inspired Approach
	6.9 Exploring the Stable Distributions Numerically
	6.10 RG-Inspired Approach for Extreme Value Distributions
	6.11 Summary
	6.12 Exercises
7 Anomalous Diffusion
	7.1 Continuous Time Random Walks
	7.2 Lévy Flights: When the Variance Diverges
	7.3 Propagator for Anomalous Diffusion
	7.4 Back to Normal Diffusion
	7.5 Ergodicity Breaking: When the Time Average and the Ensemble Average Give Different Results
	7.6 Summary
	7.7 Exercises
8 Random Matrix Theory
	8.1 Level Repulsion between Eigenvalues: The Birth of RMT
	8.2 Wigner’s Semicircle Law for the Distribution of Eigenvalues
	8.3 Joint Probability Distribution of Eigenvalues
	8.4 Ensembles of Non-Hermitian Matrices and the Circular Law
	8.5 Summary
	8.6 Exercises
9 Percolation Theory
	9.1 Percolation and Emergent Phenomena
	9.2 Percolation on Trees – and the Power of Recursion
	9.3 Percolation Correlation Length and the Size of the Largest Cluster
	9.4 Using Percolation Theory to Study Random Resistor Networks
	9.5 Summary
	9.6 Exercises
Appendix A Review of Basic Probability Concepts and Common Distributions
	A.1 Some Important Distributions
	A.2 Central Limit Theorem
Appendix B A Brief Linear Algebra Reminder, and Some Gaussian Integrals
	B.1 Basic Linear Algebra Facts
	B.2 Gaussian Integrals
Appendix C Contour Integration and Fourier Transform Refresher
	C.1 Contour Integrals and the Residue Theorem
	C.2 Fourier Transforms
Appendix D Review of Newtonian Mechanics, Basic Statistical Mechanics, and Hessians
	D.1 Basic Results in Classical Mechanics
	D.2 The Boltzmann Distribution and the Partition Function
	D.3 Hessians
Appendix E Minimizing Functionals, the Divergence Theorem, and Saddle-Point Approximations
	E.1 Functional Derivatives
	E.2 Lagrange Multipliers
	E.3 The Divergence Theorem (Gauss’s Law)
	E.4 Saddle-Point Approximations
Appendix F Notation, Notation…
References
Index