# Learning seminar

## Self-avoiding walk, spin systems, and renormalisation

Wednesdays 14:00-16:00 (seminar room, third floor, lab building west)**Schedule**(subject to large fluctuations)

- Jan 24
- Equilibrium statistical mechanics
- Topics: Ensembles, relation to quantum theory, spin systems
- Notes
- Jan 31
- Gibbs states for the classical Ising model
- Topics: Definition, reduction of existence proof
- Notes (updated Feb 1)
- References: GHM99, FV17
- Feb 7
- cancelled
- Feb 14
- Correlation inequalities and existence of Gibbs states
- Topics: Holley/FKG inequalities via coupling, existence, extremality and other properties
- Notes
- References: GHM99, FV17
- Feb 21
- cancelled
- Feb 28
- Very high temperature regime
- Topics: high-temperature expansion, uniqueness of Gibbs state, cluster expansion, analyticity of pressure
- Mar 7
- Very low temperature regime
- Topics: Peierls argument/low-temperature expansion, non-uniqueness of Gibbs states, exponential decay of correlations
- Mar 14
- More applications of cluster expansion
- Topics: Ising model in a large magnetic field, Mayer expansion, virial expansion for lattice gas
- Mar 21
- Critical phenomena
- Topics: predicted behaviour, models of walks, mean-field Ising model; time-permitting: Kac-Siegert representation, spherical model
- Mar 28
- Representations of spin systems
- Topics: random walk representation; time-permitting: triviality in high dimensions
- Apr 4
- Representations of walks
- Topics: Grassmann integrals, SAW/WSAW integral representations
- Apr 11
- Renormalisation group
- Topics: basic idea, hierarchical free field, hierarchical RG

**Other possible topics**

1-dimensional long-range Ising model, Lee-Yang circle theorem, models with continuous symmetry (Mermin-Wagner theorem and infrared bound), random current representation (application: continuity of Ising phase transition)

**References**

Statistical mechanics, spin systems, Gibbs measure, SAW:

- FV17: S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, 2017.
- Bau16: R. Bauerschmidt, Ferromagnetic spin systems, 2016.
- Geo11: H.-O. Georgii, Gibbs measures and phase transitions, 2011.
- GHM99: H.-O. Georgii, O. Häggström, and C. Maes, The random geometry of equilibrium phases, 1999.
- BDGS10: R. Bauerschmidt, H. Duminil-Copin, J. Goodman, and G. Slade, Lectures on self-avoiding walks, 2010.
- PS16: R. Peled, Y. Spinka, Lecture Notes on the Spin and Loop O(n) models, 2016.

- D. Brydges, A short course on cluster expansions, 1986.
- D. Ueltschi, Cluster expansions and correlation functions, 2005.

- D. Brydges, Lectures on the Renormalisation Group, 2009.
- R. Bauerschmidt, D. Brydges, and G. Slade, Renormalisation group analysis of 4D spin models and self-avoiding walk, 2016.

- R. Fernández, J. Fröhlich, and A. Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory, 1992.
- D. Brydges, J. Imbrie, and G. Slade, Functional integral representations for self-avoiding walk, 2009.
- H. Duminil-Copin, Graphical representations of lattice spin models, 2016.
- H. Duminil-Copin, Lectures on the Ising and Potts models on the hypercubic lattice, 2017.