Arctic boundaries of the ice model on three-bundle domains

摘要

In this paper we consider the six-vertex model at ice point on an arbitrary three-bundle domain, which is a generalization of the domain-wall ice model on the square (or, equivalently, of a uniformly random alternating sign matrix). We show that this model exhibits the arctic boundary phenomenon,whose boundary is given by a union of explicit algebraic curves. This was originally predicted by Colomo and Sportiello (J Stat Phys 164:1488-1523,2016) as one of the initial applications of a general heuristic that they introduced for locating arctic boundaries, called the (geometric) tangent method. Our proof uses a probabilistic analysis of non-crossing directed path ensembles to provide a mathematical justification of their tangent method heuristic in this case, which might be of independent interest.