Introduction to a Renormalisation Group Method

Description

This is a primer on a mathematically rigorous renormalisation group theory, presenting mathematical techniques fundamental to renormalisation group analysis such as Gaussian integration, perturbative renormalisation and the stable manifold theorem. It also provides an overview of fundamental models in statistical mechanics with critical behaviour, including the Ising and φ4 models and the self-avoiding walk. The book begins with critical behaviour and its basic discussion in statistical mechanics models, and subsequently explores perturbative and non-perturbative analysis in the renormalisation group. Lastly it discusses the relation of these topics to the self-avoiding walk and supersymmetry. Including exercises in each chapter to help readers deepen their understanding, it is a valuable resource for mathematicians and mathematical physicists wanting to learn renormalisation group theory. Front Matter ....Pages i-xii Front Matter ....Pages 1-1 Spin Systems (Roland Bauerschmidt, David C. Brydges, Gordon Slade)....Pages 3-28 Gaussian Fields (Roland Bauerschmidt, David C. Brydges, Gordon Slade)....Pages 29-36 Finite-Range Decomposition (Roland Bauerschmidt, David C. Brydges, Gordon Slade)....Pages 37-52 The Hierarchical Model (Roland Bauerschmidt, David C. Brydges, Gordon Slade)....Pages 53-65 Front Matter ....Pages 67-67 The Renormalisation Group Map (Roland Bauerschmidt, David C. Brydges, Gordon Slade)....Pages 69-88 Flow Equations and Main Result (Roland Bauerschmidt, David C. Brydges, Gordon Slade)....Pages 89-104 Front Matter ....Pages 105-105 The Tz-Seminorm (Roland Bauerschmidt, David C. Brydges, Gordon Slade)....Pages 107-121 Global Flow: Proof of Theorem 4.2.1 (Roland Bauerschmidt, David C. Brydges, Gordon Slade)....Pages 123-138 Nonperturbative Contribution to \(\varPhi _+^U\): Proof of Theorem 8.2.5 (Roland Bauerschmidt, David C. Brydges, Gordon Slade)....Pages 139-149 Bounds on \(\varPhi _+^K\): Proof of Theorem 8.2.4 (Roland Bauerschmidt, David C. Brydges, Gordon Slade)....Pages 151-179 Front Matter ....Pages 181-181 Self-Avoiding Walk and Supersymmetry (Roland Bauerschmidt, David C. Brydges, Gordon Slade)....Pages 183-210 Front Matter ....Pages 211-211 Appendix A: Extension to Euclidean Models (Roland Bauerschmidt, David C. Brydges, Gordon Slade)....Pages 213-233 Appendix B: Solutions to Exercises (Roland Bauerschmidt, David C. Brydges, Gordon Slade)....Pages 235-269 Back Matter ....Pages 271-283