NO LECTURE ON 6 NOV // EXTRA LECTURE ON 11 DEC
extra lecture on Dec 11th
new lecture notes uploaded 2024-11-10
for any info request related to the course contact me at [email protected]
After a brief probabilistic introduction (covering classical topics like laws of large numbers, central limit theorem, basic large deviation theorems) I will present the mathematical aspects of the problem of phase transitions in classical statistical physics. The main topics covered are:
Curie-Weiss mean field theory (including rigorous treatment of critical fluctuations).
The Ising model on Z^d; thermodynamic limits.
Analyticity of thermodynamic functions at high temperature and/or low density: Kirkwood-Salsburg equations, Lee-Yang theorem.
Phase transition in the Ising model: The Peierls argument.
Correlation inequalities: Griffiths-Kelly-Sherman; Fortuin-Kateleyn-Ginibre.
Models with continuous symmetries: The classical Heisenberg (or, O(N)) model.
The Mermin-Wagner theorem: no long range order at positive temperature in d=2
The Fröhlich-Simon-Spencer theorem: reflection positivity and infrared bounds, long range order at low positive temperatures in d>2.
Prerequisites:
Basic probability theory
Basic analysis
Open mind
PROBABILITY WARM-UP (WLLN, CLT, LDP); GIBBS WIEGHTS AND CANONICAL DISTRIBUTION; THE ISING MODEL. (last modified: 2024-10-16)
THE CURIE-WEISS MEAN FIELD MODEL; THE ISING MODEL ON Z^d: THERMODYNAMIC LIMIT (last modified: 2024-11-10)
THE ISING MODEL ON Z^d (ctd): ANALITICITY: (1) KIRKWOOD-SALSBURG EQUATIONS & (2) LEE-YANG THEOREM (last modified: 2024-11-10)
THE ISING MODEL ON Z^d (ctd): PEIERLS'S CONTOUR ARGUMENT AND PHASE TRANSITION; CORRELATION INEQUALITIES (last modified: 2024-11-10)
THE CLASSICAL HEISENBERG MODEL (A.K.A. THE $O(\nu)$ MODEL): NO LRO IN 2-D (MERMIN-WAGNER), LRO IN 3-D (FRÖHLICH-SIMON-SPENCER) (last modified: 2024-12-08)