Topics in Contemporary Probability and Its ApplicationsProbability theory has grown from a modest study of simple games of change to a subject with application in almost every branch of knowledge and science. In this exciting book, a number of distinguished probabilists discuss their current work and applications in an easily understood manner. Chapters show that new directions in probability have been suggested by the application of probability to other fields and other disciplines of mathematics. The study of polymer chains in chemistry led to the study of self-avoiding random walks; the study of the Ising model in physics and models for epidemics in biology led to the study of the probability theory of interacting particle systems. The stochastic calculus has allowed probabilists to solve problems in classical analysis, in theory of investment, and in engineering. The mathematical formulation of game theory has led to new insights into decisions under uncertainty. These new developments in probability are vividly illustrated throughout the book. |
Contents
Uniform Random Spanning Trees | 1 |
Simple and SelfAvoiding | 55 |
Some Connections Between Brownian Motion and Analysis | 75 |
Can You Feel the Shape of a Manifold With Brownian Motion? | 89 |
Rick Durrett 103 GreenbergHastings model | 103 |
some larger experiments David Griffeath | 113 |
References | 114 |
Problems For Students of Probability | 117 |
Random Graphs in Ecology Joel E Cohen 225 233 1 Historical background | 233 |
Characteristics of directed graphs useful for a measurement and theory 3 Models | 239 |
Connections | 241 |
Basic properties of the heterogeneous cascade model for finites | 243 |
Limit theory of the linear cascade model for large | 247 |
Dynamics of food webs References 233 240 241 243 247 | 251 |
How Many Times Should You Shuffle a Deck of Cards? Brad Mann | 261 |
What is a shuffle really? | 262 |
Systems and deterministic case | 119 |
Measures | 128 |
Random systems | 130 |
Proofs of ergodicity and fast convergence | 134 |
Percolation systems | 139 |
Nonergodicity and slow convergence | 140 |
Standard votings | 147 |
Onedimensional conservators | 149 |
Chaos approximation | 153 |
References | 154 |
MetropolisType Monte Carlo Simulation Algorithms and Simulated Annealing Basilis Gidas | 159 |
Introduction | 160 |
Metropolistype Monte Carlo simulation algorithms | 217 |
Simulated annealing | 225 |
159 | 230 |
The riffle shuffle | 265 |
How far away from randomness? 265 | 268 |
Rising sequences | 269 |
ashuffles | 270 |
Virtues of the ashuffle | 271 |
Putting it all together 9 The inverse shuffle | 274 |
Another approach to sufficient shuffling | 279 |
Stochastic Games and Operators | 291 |
269 | 319 |
The Bandit Model For Decision Processes | 321 |
Three Bewitching Paradoxes | 355 |
| 371 | |
| 372 | |
Common terms and phrases
a-shuffle algorithms assume asymptotic attractors b₁ bandit model bounded Brownian motion calculate called cards cascade model choose conditional probability convergence corresponding deck defined denote density deterministic distribution door dynamics edge eigenvalue Equation ergodic example Exercise expected number Figure finite food webs function Gittins index given graph hence infinite initial integer interleaving irreducible Lemma Marilyn vos Savant Markov chain mathematical matrix game measure modified 1-armed bandit modified bandit n-armed neighbors operator optimal P₁ payoff permutation play player problem proof Prove random system random variable random walk reach G result Riemannian Riemannian manifold riffle shuffle rising sequences satisfies simple random walk simulated annealing space spanning trees species stochastic game strategy subsection switch t₁ Theorem theory transition probability uniform uniform spanning tree v(wo v₁ vertex vertices voltage zero