Author | Madras, Neal. author |
---|---|
Title | The Self-Avoiding Walk [electronic resource] / by Neal Madras, Gordon Slade |
Imprint | Boston, MA : Birkhรคuser Boston, 1996 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-4132-4 |
Descript | XIV, 427 p. online resource |
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 Kestenโs 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 Kestenโs 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 โmyopicโ or โtrueโ self-avoiding walk -- A Random walk -- B Proof of the renewal theorem -- C Tables of exact enumerations -- Notation