Lectures on Self-Avoiding Walks

Abstract

These lecture notes provide a rapid introduction to a number of rigorous results on self-avoiding walks, with emphasis on the critical behaviour. Following an introductory overview of the central problems, an account is given of the Hammersley--Welsh bound on the number of self-avoiding walks and its consequences for the growth rates of bridges and self-avoiding polygons. A detailed proof that the connective constant on the hexagonal lattice equals $\sqrt{2+\sqrt{2}}$ is then provided. The lace expansion for self-avoiding walks is described, and its use in understanding the critical behaviour in dimensions $d>4$ is discussed. Functional integral representations of the self-avoiding walk model are discussed and developed, and their use in a renormalisation group analysis in dimension 4 is sketched. Problems and solutions from tutorials are included.

73 pages, 15 figures. Lecture notes for course given at XIV Brazilian School of Probability and Clay Mathematics Institute 2010 Summer School "Probability and Statistical Physics in Two and more Dimensions", Búzios - Rio de Janeiro (BR), 2-7 August 2010. Final version to appear on Clay Mathematics Proceedings 15 (2012)

References in corpus (9)

• Critical Exponents, Hyperscaling and Universal Amplitude Ratios for Two- and Three-Dimensional Self-Avoiding Walks
• Accurate estimate of the critical exponent $\nu$ for self-avoiding walks via a fast implementation of the pivot algorithm
• A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice
• High-precision determination of the critical exponent gamma for self-avoiding walks
• Finite range Decomposition of Gaussian Processes
• Conformal Invariance and Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk - Monte Carlo Tests
• Holomorphic Parafermions in the Potts model and SLE
• Functional integral representations for self-avoiding walk
• Long-range self-avoiding walk converges to alpha-stable processes

Cited by in corpus (17)

• Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis
• Self-avoiding walk is sub-ballistic
• Calculation of the connective constant for self-avoiding walks via the pivot algorithm
• Critical two-point function of the 4-dimensional weakly self-avoiding walk
• Percolation since Saint-Flour
• Counting self-avoiding walks
• Discrete stress-energy tensor in the loop O(n) model
• Positive speed self-avoiding walks on graphs with more than one end
• Self-avoiding walks and polygons -- an overview
• Monte Carlo study of four-dimensional self-avoiding walks of up to one billion steps
• On self-avoiding polygons and walks: counting, joining and closing
• Renormalisation group analysis of 4D spin models and self-avoiding walk
• Bounds on connective constants of regular graphs
• Self-avoiding walks and the Fisher transformation
• Strict inequalities for connective constants of transitive graphs
• An upper bound on the number of self-avoiding polygons via joining
• Three theorems in discrete random geometry