The Arctic Circle revisited

Abstract

The problem of limit shapes in the six-vertex model with domain wall boundary conditions is addressed by considering a specially tailored bulk correlation function, the emptiness formation probability. A closed expression of this correlation function is given, both in terms of certain determinant and multiple integral, which allows for a systematic treatment of the limit shapes of the model for full range of values of vertex weights. Specifically, we show that for vertex weights corresponding to the free-fermion line on the phase diagram, the emptiness formation probability is related to a one-matrix model with a triple logarithmic singularity, or Triple Penner model. The saddle-point analysis of this model leads to the Arctic Circle Theorem, and its generalization to the Arctic Ellipses, known previously from domino tilings.