| 概要 | We prove that Gibbs states of the  -color nearest neighbour Potts model on  are convex combinations of the pure phases; in particular, they are all translation invariant.
In the Ising case  translation invariance of Gibbs states is the celebrated Aizenman-Higuchi theorem. It was established directly for infinite volume states. Our approach is different and it relies on an intuitive idea that interfaces in two dimensions are delocalized. Such idea was implemented for a large class of very low temperature lattice models by Dobrushin and Shlosman. Moderately low temperatures require more care, and for the moment we can treat only the nearest neighbour case which offers a natural notion of microscopic interfaces. Following a recent work by Coquille and Velenik we consider Potts models in large finite boxes with arbitrary boundary conditions, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at all sub-critical temperatures.
Joint work with Loren Coquille, Hugo Duminil-Copin and Yvan Velenik |