The Self-Avoiding Walk

2013
ISBN: 978-1-4614-6024-4
Verlag: Springer

Buch, Englisch, 427 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 6672 g

Reihe: Modern Birkhäuser Classics

The self-avoiding walk is a mathematical model that has important applications in statistical mechanics and polymer science. In spite of its simple definition—a path on a lattice that does not visit the same site more than once—it is difficult to analyze mathematically. The Self-Avoiding Walk provides the first unified account of the known rigorous results for the self-avoiding walk, with particular emphasis on its critical behavior. Its goals are to give an account of the current mathematical understanding of the model, to indicate some of the applications of the concept in physics and in chemistry, and to give an introduction to some of the nonrigorous methods used in those fields. 

 

Topics covered in the book include: the lace expansion 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; a new 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; analysis of Monte Carlo methods that have been used to study the self-avoiding walk; the role of the self-avoiding walk in physical and chemical applications. Methods from combinatorics, probability theory, analysis, and mathematical physics play important roles. The book is highly accessible to both professionals and graduate students in mathematics, physics, and chemistry.

Zielgruppe

Graduate

Autoren/Hrsg.

Slade, Gordon

Madras, Neal

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Stochastische Prozesse
• Naturwissenschaften Chemie Physikalische Chemie Quantenchemie, Theoretische Chemie
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

Preface.- Introduction.- Scaling, polymers and spins.- Some combinatorial bounds.- Decay of the two-point function.- The lace expansion.- Above four dimensions.- Pattern theorems.- Polygons, slabs, bridges and knots.- Analysis of Monte Carlo methods.- Related Topics.- Random walk.- Proof of the renewal theorem.- Tables of exact enumerations.- Bibliography.- Notation.- Index.