Correlation nequalities in statistical mechanics
Meeting Room 3, CMS
I present an overview of work done by mathematical physicists, mostly in the 1970s and 1980s, that proves a variety of increasingly sophisticated inequalities for the correlation functions of ferromagnetic Ising and related models. Many of these constructions are ingenious but ad hoc, and cry out for systematization and extension by combinatorialists. I will discuss in particular the duplicate-variables method, the random-current method (and graphical predecessors), and the random-walk method. Moreover, there exist many plausible (and useful) conjectured correlation inequalities that we have no idea how to prove. I hope to interest combinatorialists and probabilists in attacking some of these unsolved problems.
- http://www.springerlink.com/content/g55158u007723646/fulltext.pdf - G.S. Sylvester, J. Statist. Phys. 15 (1976), 327--341
- http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103921614 - M. Aizenman, Comm. Math. Phys. 86 (1982), 1--48