This book presents fundamental relations between random walks on graphs and field theories of mathematical physics. Such relations have been explored for several decades and remain a rapidly developing research area in probability theory. The main objects of study include Markov loops, spanning f
Random Walks, Random Fields, and Disordered Systems
✍ Scribed by Anton Bovier, David Brydges, Amin Coja-Oghlan, Dmitry Ioffe, Gregory F. Lawler (auth.), Marek Biskup, Jiří Černý, Roman Kotecký (eds.)
- Publisher
- Springer International Publishing
- Year
- 2015
- Tongue
- English
- Leaves
- 254
- Series
- Lecture Notes in Mathematics 2144
- Edition
- 1
- Category
- Library
No coin nor oath required. For personal study only.
✦ Synopsis
Focusing on the mathematics that lies at the intersection of probability theory, statistical physics, combinatorics and computer science, this volume collects together lecture notes on recent developments in the area. The common ground of these subjects is perhaps best described by the three terms in the title: Random Walks, Random Fields and Disordered Systems. The specific topics covered include a study of Branching Brownian Motion from the perspective of disordered (spin-glass) systems, a detailed analysis of weakly self-avoiding random walks in four spatial dimensions via methods of field theory and the renormalization group, a study of phase transitions in disordered discrete structures using a rigorous version of the cavity method, a survey of recent work on interacting polymers in the ballisticity regime and, finally, a treatise on two-dimensional loop-soup models and their connection to conformally invariant systems and the Gaussian Free Field. The notes are aimed at early graduate students with a modest background in probability and mathematical physics, although they could also be enjoyed by seasoned researchers interested in learning about recent advances in the above fields.
✦ Table of Contents
Front Matter....Pages i-xiii
From Spin Glasses to Branching Brownian Motion—and Back?....Pages 1-64
The Renormalization Group and Self-avoiding Walk....Pages 65-116
Phase Transitions in Discrete Structures....Pages 117-146
Multidimensional Random Polymers: A Renewal Approach....Pages 147-210
Loop Measures and the Gaussian Free Field....Pages 211-235
Back Matter....Pages 237-242
✦ Subjects
Mathematical Physics; Phase Transitions and Multiphase Systems; Probability and Statistics in Computer Science; Discrete Mathematics; Probability Theory and Stochastic Processes
📜 SIMILAR VOLUMES
Volume 1 can be read without reference to Volume 2. In Volume 1, I expect of the reader only a modest facility in classical analysis, including the theory of functions of a complex variable up to contour integration. Those elements of probability theory which are needed are introduced in Chapter 1 o