Steklov Institute of Mathematics at St.Petersburg

PREPRINT 25/2006

THE ARCTIC CIRCLE REVISITED

This preprint was accepted December 25, 2006

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.