The self-avoiding walk

Neal Madras, Gordon Slade

The self-avoiding walk is a mathematical model with important applications in statistical mechanics and polymer science. This text provides a unified account of the rigorous results for the self-avoiding walk, focusing on its critical behaviour. Topics discussed include: the lace explosion and its application to the self-avoiding walk in more than four dimensions where most issues are now resolved; an introduction to the nonrigorous scaling theory; classical work of Hammersley and others; an exposition of Kesten's pattern theorem and its consequences; a discussion of the decay of the two-point function and its relation to probabilistic renewal theory; and the role of the self-avoiding walk in physical and chemical applications.

「Nielsen BookData」より

The self-avoiding walk is a mathematical model with important applications in statistical mechanics and polymer science. This text provides a unified account of the rigorous results for the self-avoiding walk, focusing on its critical behaviour. Topics discussed include: the lace explosion and its application to the self-avoiding walk in more than four dimensions where most issues are now resolved; an introduction to the nonrigorous scaling theory; classical work of Hammersley and others; an exposition of Kesten's pattern theorem and its consequences; a discussion of the decay of the two-point function and its relation to probabilistic renewal theory; and the role of the self-avoiding walk in physical and chemical applications.

「Nielsen BookData」より

[目次]

  • 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.

「Nielsen BookData」より

この本の情報

書名 The self-avoiding walk
著作者等 Madras, Neal Noah
Slade, Gordon Douglas
Slade Gordon
Madras Neal
シリーズ名 Probability and its applications
出版元 Birkhäuser
刊行年月 c1996
版表示 New ed
ページ数 xiv, 425 p.
大きさ 24 cm
ISBN 0817638911
3764338911
0817635890
NCID BA28771838
※クリックでCiNii Booksを表示
言語 英語
出版国 アメリカ合衆国
この本を: 
このエントリーをはてなブックマークに追加

このページを印刷

外部サイトで検索

この本と繋がる本を検索

ウィキペディアから連想