Statistical Mechanics of Disordered Systems: A Mathematical Perspective

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Derrida's generalised random energy models; The SK models and the Parisi solution; Hopfield models; The number partitioning problem; Bibliography; Index of notation; Index.

Professor Reimer Kühn

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This self-contained book is a graduate-level introduction for mathematicians and for physicists interested in the mathematical foundations of the field, and can be used as a textbook for a two-semester course on mathematical statistical mechanics. It assumes only basic knowledge of classical physics and, on the mathematics side, a good working knowledge of graduate-level probability theory.

The book starts with a concise introduction to statistical mechanics, proceeds to disordered lattice spin systems, and concludes with a presentation of the latest developments in the mathematical understanding of mean-field spin glass models.

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In particular, progress towards a rigorous understanding of the replica symmetry-breaking solutions of the Sherrington-Kirkpatrick spin glass models, due to Guerra, Aizenman-Sims-Starr and Talagrand, is reviewed in some detail. Passar bra ihop. Gaussian Processes on Trees Anton Bovier.

Miscellaneous back pages, Comm. Math. Phys., Volume , Number 2 ()

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  • Create citation alert. Buy this article in print. When we called for contributions to this special issue we stressed the fact that we were interested in having the topic interpreted broadly.

    Passar bra ihop

    We asked for contributions ranging from equilibrium and dynamical studies of spin glasses, glassy behaviour in amorphous materials and low temperature physics, to applications in non-conventional areas such as error-correcting codes, image analysis and reconstruction, optimization and algorithms based on statistical mechanical ideas. This was because we believe that we have arrived at a very exciting moment for the development of this multidisciplinary approach, and that this issue should bear witness to, and summarize, such an exciting situation.

    Even a cursory look at the index of this issue shows, we believe, that our hopes have been completely fulfilled; we have a large variety of papers giving new insights into the whole range of fields. Our hope is that it will play a double role. On the one hand it will carry, as good journals always should, a number of good physics papers containing important results.

    On the other it should be a good summary of the state of the art for some time. The larger section of the issue is about slow dynamics. This is understandable, since slow dynamics is such a ubiquitous phenomenon. Here recent progress deals with glasses, spin glasses and far more general situations.

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    Modifications of the celebrated fluctuation--dissipation theorem also play a crucial role. Finite dimensional systems mainly spin glasses are attracting a lot of attention since their behaviour is not yet well defined from a theoretical point of view. Here we have papers about Ising, Heisenberg and Potts spin glasses, together with the discussion of different types of disorder. Even if the mean field theory of spin glasses is well understood, important questions about, for example, the value of the complexity and the detailed nature of the solution of the model are still open, and they are discussed in this issue.

    This period has also seen very rich developments in rigorous results concerning complex disordered systems. We present some new rigorous results in this issue.

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    As we have said before, we have focused on the strong paradigmatic and interdisciplinary nature of the recent developments in the subject.