# Prague Phase Transitions

Summer School

A three-week Graduate Course on various topics in Mathematical Statistical Physics will be held in Prague, for the second time. Each lecturer will present a series of 2--3 talks, on Mathematics and Physics of Phase Transitions, accessible for graduate, post-doc, as well as advanced undergraduate students.

The lectures will take place at the Center for Theoretical Study and the Faculty of Mathematics and Physics, Charles University. In addition to 2 two-hour lectures each day, talks and seminars by post-docs and students are planned. Some additional informal activities in Prague and its surroundings will be organized in the remaining time.

Each lecturer is going to present a minicourse consisting of up to 3 talks (a talk is be default something between 3/2-2 hours).

**The tentative program:**

**April 6--9**

**C. Borgs:** First order transitions and finite-size scaling

**J. T. Chayes:** Percolation and the Random Cluster Model [A course on percolation pdf]

**J.-D. Deuschel:** Some Aspects of Continuous Gradient Systems

**April 14--17**

**G. Ben-Arous:** Spin Glasses: Statics and Dynamics

**C. Gruber:** Grassmann algebras and applications to itinerant electrons

**F. den Hollander:** Probabilistic models for polymer chains

**Y. Velenik:** Macroscopic description of phase separation.

Abstract: Thermodynamics states that the macroscopic geometry of a system in the phase separation regime can be obtained through a minimization of the total surface free energy. We discuss how this variational problem can be derived from the first principles of statistical mechanics in the case of the 2D Ising model.

**April 20--24**

**F. Dunlop:** Cluster Expansions around the Gaussian Capillary Waves Model [Abstract]

**C. Maes:** Many-particle systems with unbounded interactions A short overview of the standard Gibbsian set-up and standard Glauber dynamics. Indication how non-Gibbsian measures can arise. Discussion of concepts like almost Gibbsian and weakly Gibbsian measures. The picture, as of today, for the Dobrushin program on Gibbsian restorations. Dynamical analogues with an application to sandpiles. Relation with random unbounded interactions. References : 1. van Enter, Fernandez, Sokal: "the thick paper", 2. paper by Maes, Redig, van Moffaert in mp-arch

**C. Kuelske:** Metastates and Chaotic size dependence in random systems In disordered systems the large volume behavior of the finite volume Gibbs-measures with fixed realization of the disorder can be complicated in the presence of phase transitions. Some concepts to describe the asymptotic behavior in these cases are discussed and illustrated in mean-field type examples. References: 1. Newman, C.M.: Topics in Disordered Systems, Lectures in Mathematics, ETH Zurich, Birkh\"auser (1997), 2. C. Kuelske: Metastates in Disordered Mean Field Models: Random Field and Hopfield Models, J.Stat.Phys 88 Issue 5/6 (1997)

**O. Hryniv:** One-dimensional polymer models: some limit results 1D polymer models arise naturally in many areas of mathematics, from combinatorics and number theory to probability theory and statistical mechanics. We discuss some general approaches to investigating such models and apply them to study the surface tension in 2D contour models (Ising, SAW etc). References: 1. Graham, R.L.(ed.); Groetschel, Martin (ed.); Lovasz, Laszlo (ed.) Handbook of combinatorics. Vol. 1-2. Amsterdam: Elsevier (North-Holland), (1995). 2. Dobrushin, R.; Kotecky, R.; Shlosman, S. Wulff construction. A global shape from local interaction. Translations of Mathematical Monographs. 104. Providence, RI: AMS (1992). 3. R. Dobrushin , O. Hryniv: Fluctuations of the Phase Boundary in the 2D Ising Ferromagnet Comm Math Phys 189 (1997) 2, 395-445 and ESI preprint (no. 355)

**W.A. Majewski:** Quantum chaos

Unfortunately, no financial subsistence can be provided this time, except in exceptional cases (some very limited resources are available from a local grant). However, living expenses for the duration of the entire Course will be less than 600 $ (lodging will be available for about 100 $ per week, additional living costs are less than 100 $ per week).

Students interested please sign up via e-mail on the address: biskup@cucc.ruk.cuni.cz