2009/07/31 by Filippo Colomo, F. Colomo, A. G. Pronko · 1 citation
Mathematics · Physics and Astronomy · #Algebraic curve #Algebraic cycle #Algebraic number #Asymptote #Combinatorics #Geometry #Mathematical analysis #Mathematics #Parametric equation #Pure mathematics #Random Matrices and Applications #Saddle point #Scaling #Scaling limit #Stable curve #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Vertex (graph theory) #Vertex model #cond-mat.stat-mech #math-ph #math.CO #math.MP
paper · pdf · doi:10.1007/s10955-009-9902-2
published as J. Stat. Phys. 138 (2010), 662-700 · 39 pages, 2 figures; minor corrections
arxiv created 2009/09/02 · openalex publication_date 2009/12/11 · arxiv updated 2011/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The problem of the form of the `arctic' curve of the six-vertex model with domain wall boundary conditions in its disordered regime is addressed. It is well-known that in the scaling limit the model exhibits phase-separation, with regions of order and disorder sharply separated by a smooth curve, called the arctic curve. To find this curve, we study a multiple integral representation for the emptiness formation probability, a correlation function devised to detect spatial transition from order to disorder. We conjecture that the arctic curve, for arbitrary choice of the vertex weights, can be characterized by the condition of condensation of almost all roots of the corresponding saddle-point equations at the same, known, value. In explicit calculations we restrict to the disordered regime for which we have been able to compute the scaling limit of certain generating function entering the saddle-point equations. The arctic curve is obtained in parametric form and appears to be a non-algebraic curve in general; 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<q≤ 4) large alternating sign matrices. In particular, as q→ 0 the limit shape tends to a nontrivial limiting curve, given by a relatively simple equation.