Self-Avoiding Walk

• 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 randomwalk.- 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.

"An excellent introduction for graduate students and professional probabilists... The best place to find a self-contained exposition of lace expansion." -Bulletin of the AMS "As a carefully written and carefully referenced exposition of an intriguing topic...this monograph is strongly recommended." -Monatshefte Mathematik "In this book, the reader will find basically everything there is to know about rigorous mathematical results on self-avoiding walks... It is nicely written and should be read by mathematical physicists and probabilists interested in applications to natural sciences." -Belgian Mathematical Society "This is the first book on self-avoiding random walk and a very good one." -SIAM Review "An excellent book that should be on the shelf of anyone doing work at the intersection of probability and critical phenomena... The best results about the SAW can still be found here." --Annals of Probability