The renormalisation group — a mathematical perspective
IHES/Zoom, March 14-25, 2022
Roland Bauerschmidt
Outline
- Lecture 1. Spin systems and the renormalised potential
- Lecture 2. Perturbation theory and its three problems
- Lecture 3. Finite-range approach to renormalisation
- Lecture 4. Example of the 0-state Potts model and concluding remarks
Further references
- Lecture 1. [Wilson-83] for historical perspective; [LNM-19, Section 2] on Gaussian integration; [LSI-19] on the Polchinski equation
- Lecture 2. [BSI-14] for Gram bound
- Lecture 3. [PCMI-19, Lectures 4-6]; [H02-21, Section 3]
- Lecture 4. [H02-21, Section 1-2]
- [Wilson-83] K.G. Wilson: The renormalization group and critical phenomena PDF
Paper expanding on Wilson's Nobel Prize Lecture. Excellent historical perspective.
- [PCMI-09] D. Brydges: Lectures on the renormalisation group PDF
- [LNM-19] R. Bauerschmidt, D. Brydges, G. Slade: Introduction to a renormalisation group method PDF
- [LSI-19] R. Bauerschmidt, T. Bodineau: Log-Sobolev inequality for the continuum sine-Gordon model PDF
- [H02-21] R. Bauerschmidt, N. Crawford, T. Helmuth: Percolation transition for random forests in d ≥ 3 PDF
- [BSI-14] D. Brydges, G. Slade: A renormalisation group method. I. Gaussian integration and normed algebras PDF
- [BSII-14] D. Brydges, G. Slade: A renormalisation group method. II. Approximation by local polynomials PDF
Sections 4-6 introduce the finite-range renormalisation group approach in the context of the dipole gas. This provides further background for Lecture 3.
Part I contains background material, including on Gaussian integration and finite-range decomposition.
Parts II and III contain an analysis of the hierarchical |φ|4 model presented in such a way that very much parallels the Euclidean case in the sense that the latter is strictly an extension. In particular, this involves the solution to the large field problem in that context. This is complementary to the aspects focused on these lectures.
Section 2 includes a discussion of the structure of the Polchinski equation as the method to obtain log-Sobolev inequalities. This provides further background for Lecture 1.
Section 3 is a simple context for the construction of a renormalisation group map as discussed in Lecture 3-4.