The Self-Avoiding Walk

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.

Contenu

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.- 5.5.1 Lattice trees and animals.- 5.5.2 Percolation.- 5.6 Notes.- 6 Above four dimensions.- 6.1 Overview of the results.- 6.2 Convergence of the lace expansion.- 6.2.1 Preliminaries.- 6.2.2 The convergence proof.- 6.2.3 Proof of Theorem 6.1.2.- 6.3 Fractional derivatives.- 6.4 cn and the mean-square displacement.- 6.4.1 Fractional derivatives of the two-point function.- 6.4.2 Proof of Theorem 6.1.1.- 6.5 Correlation length and infrared bound.- 6.5.1 The correlation length.- 6.5.2 The infrared bound.- 6.6 Convergence to Brownian motion.- 6.6.1 The scaling limit of the endpoint.- 6.6.2 The finite-dimensional distributions.- 6.6.3 Tightness.- 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.- 9.2.1 Curve-fitting and linear regression.- 9.2.2 Autocorrelation times: statistical theory.- 9.2.3 Autocorrelation times: spectral theory and rigorous bounds.- 9.3 Static methods.- 9.3.1 Early methods: strides and biased sampling.- 9.3.2 Dimerization.- 9.3.3 Enrichment.- 9.4 Length-conserving dynamic methods.- 9.4.1 Local algorithms.- 9.4.2 The "slithering snake" algorithm.- 9.4.3 The pivot algorithm.- 9.5 Variable-length dynamic methods.- 9.5.1 The Berretti-Sokal algorithm.- 9.5.2 The join-and-cut algorithm.- 9.6 Fixed-endpoint methods.- 9.6.1 The BFACF algorithm.- 9.6.2 Nonlocal methods.- 9.7 Proofs.- 9.7.1 Autocorrelation times.- 9.7.2 Local algorithms.- 9.7.3 The pivot algorithm.- 9.7.4 Fixed-endpoint methods.- 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.