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
Jan 31
Gibbs states for the classical Ising model
Topics: Definition, reduction of existence proof
Notes (updated Feb 1)
References: GHM99, FV17
Feb 7
Feb 14
Correlation inequalities and existence of Gibbs states
Topics: Holley/FKG inequalities via coupling, existence, extremality and other properties
References: GHM99, FV17
Feb 21
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)

Statistical mechanics, spin systems, Gibbs measure, SAW: Cluster expansions (see also FV17, Bau16): Renormalisation group: References on representations of walks and spin systems (see also Bau16):
Left: two-dimensional self-avoiding walk with 1 million steps. Right: interface curve in the critical two-dimensional Ising model with Dobrushin boundary conditions.