The Lecture Notes collect seven mini-courses presented at the 5th Prague
Summer School on Mathematical Statistical Physics that took
placeduringtwoweeksofSeptember2006.Aswithprecedingschools, it was aimed
at PhD students and young postdocs. The central theme of the volume is
what could be called "mathematics of phase transitions" in diverse
contexts. Even though all courses were meant to introduce the reader to
recent progress of a particular topic of modern statis- cal physics,
attention has been paid to providing a solid grounding by carefully
developing various basic tools. One of the techniques that led, more
than two decades ago, to a
seriesofimportantoutcomesinthetheoryofphasetransitionsoflattice models
was re?ection positivity. Recently it resurfaced and was used to obtain
interesting new results in various settings. The lectures of Marek
Biskup include a thorough introduction to re?ection positivity as well
as a review of its recent applications. In addition, it contains a crash
course on lattice spin models that is useful as a background for other
lectures of the collection. Also the following two contributions concern
equilibrium statistical
physics.ThelecturesofDmitriIo?earedevotedtoastochasticgeom-
ricreformulationofclassicalaswellasquantumIsingmodels.Auni?ed
approachtotheFortuin-Kasteleynandrandomcurrentrepresentations in terms
of path integrals is presented. Statistical mechanics of directed
polymers interacting with o-
dimensionalspatiale?ectsisatopicwithvariousapplicationsinphysics and
biophysics. The lectures of Fabio Toninelli are devoted to a th- ough
discussion of the localization/delocalization transition in these
models.
