Topics discussed in this course include methods to proof of the existence of phase transitions, reflection positivity, continuous symmetry, methods based on convexity, random walk representations, correlation inequalities, and the relation of spin systems to the self-avoiding walk.
Prerequisites: measure theoretic probability
There is no general textbook. Relevant references and lecture notes will be posted here.
The precise content will depend on the interests of the audidence. A tentative schedule is:
- Week 1. Overview and goals. Mean-field theory.
- Week 2. High temperature (1). Bounds on correlations by mean field theory
- Week 3. High temperature (2). Convexity, Brascamp-Lieb inequality, Helffer-Sjoestrand representation.
- Week 4. Low temperature. Peierls argument, reflection positivity, infrared bound, and continuous symmetry breaking.
- Week 5. Expansions for high and low temperature.
- Week 6-7. Abelian spin systems. Duality. Long-range order for the XY model.
- Week 8. Random walk and random current representations. Derivation and first consequences.
- Week 9. Triviality in dimension 5 and higher.
- Week 10. Schwinger-Dyson equation and construction of the (φ^{4})_{3} field theory.
- Week 11. Gaussian free field.
- Week 12. The renormalization group in the hierarchical approximation.