These lectures provides a brief introduction to the theory of critical phenomena, including the Ising and 4 spin models, and the strictly and weakly self-avoiding walk. Then recent results in dimension 4 for the 4 model and the weakly self-avoiding walk are discussed. These results have been obtained using a rigorous renormalisation group method. Several tools that are fundamental to the renormalisation group method are subsequently introduced, including Gaussian integration, perturbative renormalisation, and the stable manifold theorem. The last third of these lectures discusses non-perturbative aspects of the implementation of a renormalisation group step. Tutorials are included.

Introduces mathematically rigorous renormalisation group theory Explores mathematically rigorous approaches Based on the lecture notes for summer school courses

Autorentext

Roland Bauerschmidt is a University Lecturer in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge. His work mainly focuses on the applications of probability theory and analysis to topics in statistical mechanics. He received his Ph.D. in Mathematics from the University of British Columbia in 2013, and was a member of the Institute for Advanced Study in Princeton at the School of Mathematics, and a postdoctoral researcher at Harvard University. He received the IUPAP Young Scientist Prize in 2015. David C. Brydges is a Professor Emeritus of Mathematics at the University of British Columbia. He works on quantum field theory and statistical mechanics, with an interest in functional integration and probability. He received his Ph.D. from the University of Michigan, and was a Professor at the University of Virginia. He served as the President of the International Association of Mathematics Physics in 2003-2005, and was elected a Fellow of the Royal Society of Canada in 2007. Gordon Slade is a Professor in the department of Mathematics at the University of British Columbia. He works on probability theory and statistical mechanics. He received his Ph.D. in Mathematics from the University of British Columbia, and was a Lecturer at the University of Virginia. He was also an Assistant Professor, Associate Professor and Professor at McMaster University. He has been honored with various prizes, e.g., Jeffery-Williams Prize, CRM-Fields-PIMS Prize, Coxeter-James Prize, and was elected a Fellow of the Royal Society (London), the American Mathematical Society, the Institute of Mathematical Statistics, the Fields Institute and the Royal Society of Canada.

Inhalt

• Part I Spin Systems and Critical Phenomena. - Spin Systems. - Gaussian Fields. - Finite-Range Decomposition. - The Hierarchical Model. - Part II The Renormalisation Group: Perturbative Analysis . - The Renormalisation Group Map. - Flow Equations and Main Result. - Part III The Renormalisation Group: Nonperturbative Analysis. - The Tz-Seminorm. - Global Flow: Proof of Theorem 4.2.1. - Nonperturbative Contribution to U/+: Proof of Theorem 8.2.5. - Bounds on K/+ : Proof of Theorem 8.2.4. - Part IV Self-AvoidingWalk and Supersymmetry. - Self-AvoidingWalk and Supersymmetry. - Part V Appendices . - Appendix A: Extension to Euclidean Models. - Appendix B: Solutions to Exercises.