Abstract
We discuss the Fortuin-Kasteleyn (FK) random cluster representation for Ising models with no external field and with pair interactions which need not be ferromagnetic. In the ferromagnetic case, the close connections between FK percolation and Ising spontaneous magnetization and the availability of comparison inequalities to independent percolation have been applied to certain disordered systems, such as dilute Ising ferromagnets and quantum Ising models in random environments; we review some of these applications. For non-ferromagmetic disordered systems, such as spin glasses, the state of the art is much more primitive. We discuss some of the many open problems for spin glasses and show how the FK representation leads to one small result, that there is uniqueness of the spin glass Gibbs distribution above the critical temperature of the associated ferromagnet.
Supported in part by the National Science Foundation under Grant DMS 92–09053; thanks are due the Isaac Newton Institute for Mathematical Sciences for support and hospitality; NATO for its travel support to attend this Advanced Study Institute; and C. Borgs and J. Bricmont for help with references.
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Newman, C.M. (1994). Disordered Ising Systems and Random Cluster Representations. In: Grimmett, G. (eds) Probability and Phase Transition. NATO ASI Series, vol 420. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8326-8_15
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DOI: https://doi.org/10.1007/978-94-015-8326-8_15
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