00-278 Raphael Cerf, Agoston Pisztora

Phase coexistence in Ising, Potts and percolation models (715K, uuencoded gzipped Postscript file) Jun 27, 00

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Abstract. We study phase separation and phase coexistence phenomena in Ising, Potts and random cluster models in dimensions $d\ge 3$. The simultaneous occurrence of several phases is typical for systems with appropriately arranged (mixed) boundary conditions and for systems conditioned on certain large deviation events. These phases define a partition of the space. Our main results are large deviations principles for (the shape of) the {\it empirical phase partition} emerging in these models. More specifically, we establish a general large deviation principle for the partition induced by large (macroscopic) clusters in the Fortuin--Kasteleyn model and transfer it to the Ising--Potts model where we obtain a large deviation principle for the natural partition corresponding to the various phases. The rate function turns out to be the total surface energy (associated with the surface tension of the model and with boundary conditions) which can be naturally assigned to each reasonable partition. These LDP-s imply a weak law of large numbers: the law of the phase partition is asymptotically determined by an appropriate variational problem. More precisely, the phase partition will be close to some partition which is compatible with the constraints imposed on the system and which minimizes the total surface energy. A general compactness argument guarantees the existence of at least one such minimizing partition. Our results are valid for temperatures $T$ below a limit of slab-thresholds $\tchat$ conjectured to agree with the critical point $T_c$. Moreover, $T$ should be such that there exists only one translation invariant infinite volume state in the corresponding Fortuin--Kasteleyn model; a property which can fail for at most countably many values and which is conjectured to be true for every~$T$.

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