The 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.- 5.5.1 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.