# The arctic curve of the domain-wall six-vertex model - Mathematical Physics

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Abstract: The problem of the form of the `arctic- curve of the six-vertex model withdomain wall boundary conditions in its disordered regime is addressed. It iswell-known that in the scaling limit the model exhibits phase-separation, withregions of order and disorder sharply separated by a smooth curve, called thearctic curve. To find this curve, we study a multiple integral representationfor the emptiness formation probability, a correlation function devised todetect spatial transition from order to disorder. We conjecture that the arcticcurve, for arbitrary choice of the vertex weights, can be characterized by thecondition of condensation of almost all roots of the corresponding saddle-pointequations at the same, known, value. In explicit calculations we restrict tothe disordered regime for which we have been able to compute the scaling limitof certain generating function entering the saddle-point equations. The arcticcurve is obtained in parametric form and appears to be a non-algebraic curve ingeneral; it turns into an algebraic one in the so-called root-of-unity cases.The arctic curve is also discussed in application to the limit shape of$q$-enumerated with $0

Author: ** F. Colomo, A.G. Pronko**

Source: https://arxiv.org/