Featuring over 1300 exercises and solutions, this volume serves as both a comprehensive resource for practicing probability theory and a guide for educators and students. It complements the companion text, Probability and Random Processes (4th edition), providing detailed coverage of related exercises and problems, making it an essential tool for mastering the subject.
The random-cluster model was introduced by Cees Fortuin and Piet Kasteleyn around 1969 as a unification of percolation, Ising, and Potts models, aiming to reconcile the series and parallel laws governing such systems. This innovation sparked a study in stochastic geometry, revealing intricate structures and becoming essential in addressing one of the longstanding challenges in classical statistical mechanics: modeling and analyzing ferromagnetism and its phase transitions. However, the model's significance in probability and statistical mechanics wasn't fully acknowledged until the late 1980s, primarily due to two factors. First, early publications from 1969 to 1972 emphasized combinatorial aspects, which may have obscured its broader applications. Second, many necessary geometrical arguments for the model's study were not developed until the 1980s during the so-called ‘decade of percolation.’ A pivotal moment came in 1980 with the proof concerning bond percolation on the square lattice, followed by Harry Kesten’s monograph on t-dimensional percolation. By 1986, percolation theories expanded into higher dimensions, resolving many contemporary mathematical challenges. Renewed interest in the random-cluster model as a tool for exploring Ising/Potts models emerged around 1987.
Focusing on the foundational aspects of probability and random processes, this book serves as an essential resource for first and second year mathematics undergraduates and Master's students in related disciplines. It provides a concise yet comprehensive overview, highlighting the significance of probability in various fields of study and its contemporary relevance in human endeavors.
Percolation theory is the study of an idealized random medium in two or more dimensions. The mathematical theory is mature, and continues to give rise to problems of special beauty and difficulty. Percolation is pivotal for studying more complex physical systems exhibiting phase transitions. The emphasis of this book is upon core mathematical material and the presentation of the shortest and most accessible proofs. The book is intended for graduate students and researchers in probability and mathematical physics. Almost no specialist knowledge is assumed. Much new material appears in this second edition, including: dynamic and static renormalization, strict inequalities between critical points, a sketch of the lace expansion, and several essays on related fields and applications.
"This is the first comprehensive identification guide to the 1300 species found in India, Pakistan, Nepal, Bangladesh, Bhutan, Sri Lanka and the Maldives. The text comprises a detailed identification section, discussing the differences betweeen similar species and descriptions of vocalisations, habits, habitat, breeding and distribution, and status."
A user-friendly introduction for mathematicians to some of the principal stochastic models near the interface of probability and physics.
Grimmett, Geoffrey: Percolation and disordered systems.- Kesten, Harry: Aspects of first passage percolation.