Spin Glasses: Statics and Dynamics

Over the last decade, spin glass theory has turned from a fascinating part of t- oretical physics to a ? ourishing and rapidly growing subject of probability theory as well. These developments have been triggered to a large part by the mathem- ical understanding gained on the fascinating and previously mysterious "Parisi solution" of the Sherrington-Kirkpatrick mean ? eld model of spin glasses, due to the work of Guerra, Talagrand, and others. At the same time, new aspects and applications of the methods developed there have come up. The presentvolumecollects a number of reviewsaswellas shorterarticlesby lecturers at a summer school on spin glasses that was held in July 2007 in Paris. These articles range from pedagogical introductions to state of the art papers, covering the latest developments. In their whole, they give a nice overview on the current state of the ? eld from the mathematical side. The review by Bovier and Kurkova gives a concise introduction to mean ? eld models, starting with the Curie-Weiss model and moving over the Random Energymodels up to the Parisisolutionof the Sherrington-Kirkpatrikmodel. Ben Arous and Kuptsov present a more recent view and disordered systems through the so-called local energy statistics. They emphasize that there are many ways to look at Hamiltonians of disordered systems that make appear the Random Energy model (or independent random variables) as a universal mechanism for describing certain rare events. An important tool in the analysis of spin glasses are correlation identities.

Mean Field. - A Short Course on Mean Field Spin Glasses. - REM Universality for Random Hamiltonians. - Another View on Aging in the REM. - Spin Glass Identities and the Nishimori Line. - Self-averaging Identities for Random Spin Systems. - Chaos in Mean-field Spin-glass Models. - A non Gaussian Limit Law for the Covariances of Spins in a SK Model with an External Field. - A Limit Theorem for Mean Magnetisation in the Sherrington-Kirkpatrick Model with an External Field. - Non-mean Field. - A Percolation-theoretic Approach to Spin Glass Phase Transitions. - Fluctuations in Finite-dimensional Spin-glass Dynamics. - Disordered Pinning Models. - Renewal Sequences, Disordered Potentials, and Pinning Phenomena. - A Smoothing Inequality for Hierarchical Pinning Models.