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Catalogue Information
Field name
Details
Dewey Class
519.2
Title
The Self-Avoiding Walk ([EBook] /) / by Neal Madras, Gordon Slade.
Author
Madras, Neal
Added Personal Name
Slade, Gordon
author.
Other name(s)
SpringerLink (Online service)
Publication
Boston, MA : : Birkhäuser Boston, , 1996.
Physical Details
XIV, 427 p. : online resource.
Series
Probability and Its Applications
ISBN
9781461241324
Summary Note
A self-avoiding walk is a path on a lattice that does not visit the same site more than once. In spite of this simple definition, many of the most basic questions about this model are difficult to resolve in a mathematically rigorous fashion. In particular, we do not know much about how far an n step self-avoiding walk typically travels from its starting point, or even how many such walks there are. These and other important questions about the self-avoiding walk remain unsolved in the rigorous mathematical sense, although the physics and chemistry communities have reached consensus on the answers by a variety of nonrigorous methods, including computer simulations. But there has been progress among mathematicians as well, much of it in the last decade, and the primary goal of this book is to give an account of the current state of the art as far as rigorous results are concerned. A second goal of this book is to discuss some of the applications of the self-avoiding walk in physics and chemistry, and to describe some of the nonrigorous methods used in those fields. The model originated in chem istry several decades ago as a model for long-chain polymer molecules. Since then it has become an important model in statistical physics, as it exhibits critical behaviour analogous to that occurring in the Ising model and related systems such as percolation.:
Contents note
1 Introduction -- 1.1 The basic questions -- 1.2 The connective constant -- 1.3 Generating functions -- 1.4 Critical exponents -- 1.5 The bubble condition -- 1.6 Notes -- 2 Scaling, polymers and spins -- 2.1 Scaling theory -- 2.2 Polymers -- 2.3 The N ? 0 limit -- 2.4 Notes -- 3 Some combinatorial bounds -- 3.1 The Hammersley-Welsh method -- 3.2 Self-avoiding polygons -- 3.3 Kesten’s bound on cN -- 3.4 Notes -- 4 Decay of the two-point function -- 4.1 Properties of the mass -- 4.2 Bridges and renewal theory -- 4.3 Separation of the masses -- 4.4 Ornstein-Zernike decay of GZ(0, x) -- 4.5 Notes -- 5 The lace expansion -- 5.1 Inclusion-exclusion -- 5.2 Algebraic derivation of the lace expansion -- 5.3 Example: the memory-two walk -- 5.4 Bounds on the lace expansion -- 5.5 Other models -- 5.6 Notes -- 6 Above four dimensions -- 6.1 Overview of the results -- 6.2 Convergence of the lace expansion -- 6.3 Fractional derivatives -- 6.4 cn and the mean-square displacement -- 6.5 Correlation length and infrared bound -- 6.6 Convergence to Brownian motion -- 6.7 The infinite self-avoiding walk -- 6.8 The bound on cn(0,x) -- 6.9 Notes -- 7 Pattern theorems -- 7.1 Patterns -- 7.2 Kesten’s Pattern Theorem -- 7.3 The main ratio limit theorem -- 7.4 End patterns -- 7.5 Notes -- 8 Polygons, slabs, bridges and knots -- 8.1 Bounds for the critical exponent ?sing -- 8.2 Walks with geometrical constraints -- 8.3 The infinite bridge -- 8.4 Knots in self-avoiding polygons -- 8.5 Notes -- 9 Analysis of Monte Carlo methods -- 9.1 Fundamentals and basic examples -- 9.2 Statistical considerations -- 9.3 Static methods -- 9.4 Length-conserving dynamic methods -- 9.5 Variable-length dynamic methods -- 9.6 Fixed-endpoint methods -- 9.7 Proofs -- 9.8 Notes -- 10 Related topics -- 10.1 Weak self-avoidance and the Edwards model -- 10.2 Loop-erased random walk -- 10.3 Intersections of random walks -- 10.4 The “myopic” or “true” self-avoiding walk -- A Random walk -- B Proof of the renewal theorem -- C Tables of exact enumerations -- Notation.
System details note
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site
http://dx.doi.org/10.1007/978-1-4612-4132-4
Links to Related Works
Subject References:
Mathematics
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Probabilities
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Probability theory and stochastic processes
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Authors:
Madras, Neal
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Slade, Gordon
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Corporate Authors:
SpringerLink (Online service)
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Series:
Probability and Its Applications
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Classification:
519.2
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