%0 Generic %1 Bauerschmidt2012Lectures %A Bauerschmidt, Roland %A Duminil-Copin, Hugo %A Goodman, Jesse %A Slade, Gordon %D 2012 %K review random-walks %T Lectures on Self-Avoiding Walks %U http://arxiv.org/abs/1206.2092 %X These lecture notes provide a rapid introduction to a number of rigorous results on self-avoiding walks, with emphasis on the critical behaviour. Following an introductory overview of the central problems, an account is given of the Hammersley--Welsh bound on the number of self-avoiding walks and its consequences for the growth rates of bridges and self-avoiding polygons. A detailed proof that the connective constant on the hexagonal lattice equals \$2+\sqrt2\$ is then provided. The lace expansion for self-avoiding walks is described, and its use in understanding the critical behaviour in dimensions \$d>4\$ is discussed. Functional integral representations of the self-avoiding walk model are discussed and developed, and their use in a renormalisation group analysis in dimension 4 is sketched. Problems and solutions from tutorials are included.

@misc{Bauerschmidt2012Lectures, abstract = {These lecture notes provide a rapid introduction to a number of rigorous results on self-avoiding walks, with emphasis on the critical behaviour. Following an introductory overview of the central problems, an account is given of the Hammersley--Welsh bound on the number of self-avoiding walks and its consequences for the growth rates of bridges and self-avoiding polygons. A detailed proof that the connective constant on the hexagonal lattice equals \$\sqrt{2+\sqrt{2}}\$ is then provided. The lace expansion for self-avoiding walks is described, and its use in understanding the critical behaviour in dimensions \$d\>4\$ is discussed. Functional integral representations of the self-avoiding walk model are discussed and developed, and their use in a renormalisation group analysis in dimension 4 is sketched. Problems and solutions from tutorials are included.}, added-at = {2019-06-10T14:53:09.000+0200}, archiveprefix = {arXiv}, author = {Bauerschmidt, Roland and Duminil-Copin, Hugo and Goodman, Jesse and Slade, Gordon}, biburl = {https://www.bibsonomy.org/bibtex/22f07d77bdc0e8702f6bbeaa9596ac1bb/nonancourt}, citeulike-article-id = {10782138}, citeulike-linkout-0 = {http://arxiv.org/abs/1206.2092}, citeulike-linkout-1 = {http://arxiv.org/pdf/1206.2092}, day = 11, eprint = {1206.2092}, interhash = {98bf93beb09981ec747b5532ee718eef}, intrahash = {2f07d77bdc0e8702f6bbeaa9596ac1bb}, keywords = {review random-walks}, month = jun, posted-at = {2013-01-29 15:30:35}, priority = {2}, timestamp = {2019-08-01T15:38:31.000+0200}, title = {{Lectures on Self-Avoiding Walks}}, url = {http://arxiv.org/abs/1206.2092}, year = 2012 }