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Analysis and Stochastics of Growth Processes and Interface Models$
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Peter Mörters, Roger Moser, Mathew Penrose, Hartmut Schwetlick, and Johannes Zimmer

Print publication date: 2008

Print ISBN-13: 9780199239252

Published to Oxford Scholarship Online: September 2008

DOI: 10.1093/acprof:oso/9780199239252.001.0001

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Applications of the Lace Expansion to Statistical-Mechanical Models

Applications of the Lace Expansion to Statistical-Mechanical Models

(p.123) 6 Applications of the Lace Expansion to Statistical-Mechanical Models
Analysis and Stochastics of Growth Processes and Interface Models

Akira Sakai

Oxford University Press

Synergetics is a common feature in interesting statistical-mechanical problems. One of the most important examples of synergetics is the emergence of a second-order phase transition and critical behaviour. It is rich and still far from fully understood. The reason why it is so difficult is due to the increase to infinity of the number of strongly correlated variables in the vicinity of the critical point. For example, the Ising model, which is a model for magnets, exhibits critical behaviour as the temperature comes closer to its critical value; the closer the temperature is to criticality, the more spin variables cooperate with each other to attain the global magnetization. In this regime, neither standard probability theory for independent random variables nor naive perturbation techniques work. The lace expansion, which is the topic of this article, is currently one of the few approaches to rigorous investigation of critical behaviour for various statistical-mechanical models. The chapter summarizes some of the most intriguing lace-expansion results for self-avoiding walk (SAW), percolation, and the Ising model.

Keywords:   lace expansion, phase transition, Ising model, self-avoiding walk, percolation

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