Disordered systems have been a topic of great interest in statistical physics over at least 25 years. Theoretical physics has developed astonishing tools based on intriguing mathematically heuristic tools based on what is called the 'replica trick' that have defied so far a rigorous mathematical understanding. In this series of lectures I will review some of the attempts towards a rigorous approach to disordered spin systems, concentrating on the questions of the construction and description of the low temperature phases of such systems.
These lectures fall into 4 parts:
Here I will give a short introduction to the concept of infinite volume Gibbs measures for disordered lattice spin systems as random measures. In particular I will discuss the associated notions of ``metastates'' as put forward by Ch. Newman and D. Stein.
We will begin to ask the key question: under what conditions can we have more than one Gibbs state (at low temperatures)? We will start to analyze the role played by random fluctuations by looking at the random field Ising model.
A substantial part of even the heuristic insight into disordered systems, and in particular spin glasses, comes from the study of supposedly simple `mean field models'. In the first part we look at models related to Gaussian processes on the hypercube, the so called `p-spin Sherrington-Kirkpatrick (SK) models' and the `random energy model''(REM). The latter is the only model that is completely solvable, and I will present this solution as an illustration of some of the new features one can expect in spin glass models. I will then survey results by Talagrand on the p-spin models with p>2, and discuss the difficulties to pass to the low temperature phase in the case p=2, the standard SK model.
The Hopfield model, best known in the context of neural networks, represents a different kind of mean field model which offers the advantage that one has an a priori idea of what the low temperature Gibbs states should be. This allows an analysis where random fluctuations are essentially treated as perturbations. I will explain the construction of Gibbs states and metastates in this model, and show how results from the replica method can be recovered in some cases.
The most intriguing experimental facts on spin glasses concern their dynamics. Physicists have termed the characteristic feature of these systems as ``aging''. I will explain this phenomenon in the simplest setting, the Glauber dynamics of the random energy model.
Lecture notes covering the material presented in the course are available. Other main references are:
H.-O. Georgii, Gibbs measures and phase transitions.
de Gruyter Studies in
Mathematics, 9. Walter de Gruyter & Co., Berlin-New York, 1988.
(Background on Gibbs measures).
Ch.M. Newman. Topics in disordered systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1997. (Metastates, spin glasses on the lattice)
A. Bovier and P. Picco, Mathematical aspects of spin glasses and neural networks, 243-287, Progr. Probab., 41, Birkhäuser Boston, Boston, MA, 1998. (A number of review papers, in particular on Hopfield models).
M. Aizenman, J. Wehr. Rounding effects of quenched randomness on first-order phase transitions. Comm. Math. Phys. 130 (1990), no. 3, 489-528.
J. Bricmont, A. Kupiainen. Phase transition in the 3d random field Ising model. Comm. Math. Phys. 116 (1988), no. 4, 539-572.
(The papers by Aizenman-Wehr and Bricmont-Kupiainen are the two key papers on the random field model).
M. Talagrand, Rigorous low-temperature results for the mean field p-spins interaction model, Probab Theory Relat Fields 117, 303-360 (2000).