Séminaire du 12 octobre 2009, 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).

Virginie Collette
Last modified: Mon Oct 12 14:27:37 CEST 2009