Interacting Graphical Models and Related Topics in Statistical Mechanics

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9971016This research principally concerns a class of correlated (non-independent) percolation problems that arise as graphical representations of spin-systems. These representations contain all of the information of the original spin-system and, in the best of cases, provide a definitive signal (percolation) for phase transitions or phase coexistence. The most familiar example of such a representation is the random cluster representation for the q-state Potts model. Recently, a number of new graphical representations associated with other models have emerged. In addition to their intrinsic interest, the study of these problems is important for two reasons: (i) The representations provide a powerful analytical tool with which to study the underlying statistical mechanics problem. (ii) The representations are the basis of the cluster methods, a modern class of Monte Carlo algorithms that allow for the rapid simulation of these (underlying) problems. To realize item (ii) and, for the most part item (i) as well, it is essential to achieve "the best of cases" i.e. demonstrate that the phase transition in the statistical mechanics problem is mirrored by a percolation transition in the graphical representation. In general, the PI will continue research on the development of new graphical representations and cluster algorithms with an eye towards the construction of algorithms in equilibrium statistical mechanics and other problems that are computationally complex.This research may be described as follows: The primitive mathematical descriptions of interacting systems necessarily involves a great deal of superfluous information. At best this information is simply of no importance and at worst, it can obscure the subtle effects that are the object of study. Graphical representations present an alternative description. Although the mathematical content of the description is identical to the original, the form is quite different: Much of the superfluous information is pushed into the background and the interesting features are often exhibited rather dramatically as geometric phenomena. Of further importance is that these geometric representations are the key ingredient in a class of computer algorithms (cluster algorithms) that are used in statistical physics. Until recently, only a handful of systems have admitted such descriptions but the situation has improved a great deal in the last few years. Progress will continue in expanding the scope of the applicability of these methods both in equilibrium statistical mechanics an in other area where computational complexity forbids the straightforward approach.