**Title:** Three- and Four-color Potts Models in the Plane: Vortices, Entropic Forces and Interfaces

**Speaker:** Dr. Chris Moore

**Speaker Info:** Santa Fe Institute

**Brief Description:**

**Special Note**:

**Abstract:**

Consider the set of ways to color a checkerboard with three colors, such that neighboring squares have different colors. This is the three-state antiferromagnetic Potts model. If we start with a random state and change colors one square at a time, we find that pairs of the same color act as diffusing particles. There are two kinds of pairs --- 'positive' and 'negative' --- and these attract each other with a Coulomb-like force.There is no energy gradient to drive this force; it is purely entropic, driven by the fact that the number of possible states for the lattice increases as the particles get closer together. I will explain how to derive this using an interface representation in which the state of the lattice is thought of as a surface in three dimensions, in which these particles become screw dislocations. I will give the results of numerical experiments that show this force, and discuss why this measurement is difficult to perform.

A more complex example is the four-color model on the triangular lattice. Here the surface becomes four-dimensional, and the particles acquire a vector-valued charge. I will discuss work with Mark Newman in which we show that Monte Carlo dynamics are only ergodic on lattices of certain topologies. I will also discuss how to calculate critical exponents of models like these.

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