14h00: Filippo Colomo, Università di Firenze, Italy.

The limit shape of large Alternating Sign Matrices.

The problem of the limit shape of q-enumerated (0<q<5) large Alternating Sign Matrices (ASMs) - which can also be rephrased as the problem of the Arctic curve for interacting-domino tilings of the Atzec Diamond - is addressed by exploiting the well-known correspondence with the domain-wall six vertex model. In particular we analyse the scaling limit behaviour of a multiple integral representation of the Emptiness Formation Probability in the domain-wall six-vertex model. We conjecture that the limit shape can be characterized by the condition of condensation of almost all roots of the corresponding saddle-point equations at the same, known, value. Under this assumption, we are able to derive the limit shape of q-enumerated (0<q<5) large ASMs. It is expressed in parametric form, and it appears to be a non-algebraic curve in general; it turns into an algebraic one in the so-called root-of-unity cases (which includes, among others, the famous particular cases: q = 1, 2, 3).