×

zbMATH — the first resource for mathematics

Lectures on dimers. (English) Zbl 1180.82001
Sheffield, Scott (ed.) et al., Statistical mechanics. Papers based on the presentations at the IAS/PCMI summer conference, Park City, UT, USA, July 1–21, 2007. Providence, RI: American Mathematical Society (AMS); Princeton, NJ: Institute for Advanced Study (ISBN 978-0-8218-4671-1). IAS/Park City Mathematics Series 16, 191-230 (2009).
From the text: The planar dimer model is, from one point of view, a statistical mechanical model of random 2-dimensional interfaces in \(\mathbb R^3\). In a concrete sense it is a natural generalization of the simple random walk on \(Z\). While the simple random walk and its scaling limit, Brownian motion, permeate all of probability theory and many other parts of mathematics, higher dimensional models like the dimer model are much less used or understood. Only recently have tools been developed for gaining a mathematical understanding of two-dimensional random fields. The dimer model is at the moment the most successful of these two-dimensional theories. What is remarkable is that the objects underlying the simple random walk: the Laplacian, Green’s function, and Gaussian measure, are also the fundamental tools used in the dimer model. On the other hand the study of the dimer model actually uses tools from many other areas of mathematics: we see a little bit of algebraic geometry. PDEs, analysis, and ergodic theory, at least.
Our goal in these notes is to study the planar dimer model and the associated random interface model. There has been a great deal of recent research on the dimer model but there are still many interesting open questions and new research avenues. In these lecture notes we will provide an introduction to the dimer model, leading up to results on limit shapes and fluctuations. There are a number of exercises and a few open questions at the end of each chapter. Our objective is not to give complete proofs, but we will at least attempt to indicate the ideas behind many of the proofs. Complete proofs can all be found in the papers in the bibliography.
For the entire collection see [Zbl 1165.82003].

MSC:
82-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistical mechanics
82Bxx Equilibrium statistical mechanics
97M50 Physics, astronomy, technology, engineering (aspects of mathematics education)
PDF BibTeX XML Cite
Full Text: arXiv