Three Coloring the Discrete Torus or 3-States Anti-Ferromagnetic Potts Model in Zero Temperature

Tuesday, October 7, 2014 - 2:00pm - 3:00pm
Lind 305
Ohad Feldheim (University of Minnesota, Twin Cities)
We prove that a uniformly chosen proper three coloring of Z_{2n}^d has a very rigid structure when the dimension d is sufficiently high. In particular the coloring asymptotically almost surely takes one color on almost all of either the even or the odd sub-lattice. This implies for example that one color appears on nearly half of the lattice sites.

This model is the zero temperature case of the 3-states anti-ferromagnetic Potts model, which has been studied extensively in statistical mechanics. The result improves an independent bound due to Galvin, Kahn, Randall and Sorkin. The proof, however, is quite different: using combinatorial methods which follow an algebraic-topological intuition, results of Peled about homomorphism height functions are extended to a new setting.

Joint work with Ron Peled.