Filippo Colomo

The role of orthogonal polynomials in the six-vertex model and its combinatorial applications ∗

The Hankel determinant representations for the partition function and boundary correlation functions of the six-vertex model with domain wall boundary conditions are investigated by the methods of orthogonal polynomial theory. For specific values of the parameters of the model, corresponding to 1-, 2- and 3-enumerations of alternating sign matrices (ASMs), these polynomials specialize to classical ones (continuous Hahn, Meixner–Pollaczek and continuous dual Hahn, respectively). As a consequence, a unified and simplified treatment of ASM enumerations turns out to be possible, leading also to some new results such as the refined 3-enumerations of ASMs. Furthermore, the use of orthogonal polynomials allows us to express, for generic values of the parameters of the model, the partition function of the (partially) inhomogeneous model in terms of the one-point boundary correlation functions of the homogeneous one.

Arctic Curves of the Six-Vertex Model on Generic Domains: The Tangent Method

We revisit the problem of determining the Arctic curve in the six-vertex model with domain wall boundary conditions. We describe an alternative method, by which we recover the previously conjectured analytic expression in the square domain. We adapt the method to work for a large class of domains, and for other models exhibiting limit shape phenomena. We study in detail some examples, and derive, in particular, the Arctic curve of the six-vertex model in a triangoloid domain at the ice point.

Thermodynamics of the Six-Vertex Model in an L-Shaped Domain

We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall boundary conditions. For free-fermion vertex weights the partition function can be expressed in terms of some Hankel determinant, or equivalently as a Coulomb gas with discrete measure and a non-polynomial potential with two hard walls. We use Coulomb gas methods to study the partition function in the thermodynamic limit. We obtain the free energy of the six-vertex model as a function of the parameters describing the geometry of the scaled L-shaped domain. Under variations of these parameters the system undergoes a third-order phase transition. The result can also be considered in the context of dimer models, for the perfect matchings of the Aztec diamond graph with a cut-off corner.

On some representations of the six vertex model partition function

Abstract The partition function of the six-vertex model on the finite lattice with domain wall boundary conditions is considered. Starting from Hankel determinant representation, some alternative representations for the partition function are given. It is argued that one of these representations can be rephrased in the language of the angular quantization method applied to certain fermionic model.

Generalized emptiness formation probability in the six-vertex model

In the six-vertex model with domain wall boundary conditions, the emptiness formation probability is the probability that a rectangular region in the top left corner of the lattice is frozen. We generalize this notion to the case where the frozen region has the shape of a generic Young diagram. We derive here a multiple integral representation for this correlation function.

SOLVING THE FRUSTRATED SPHERICAL MODEL WITH Q-POLYNOMIALS

We analyse the spherical model with frustration induced by an external gauge field. This has been recently mapped in infinite dimensions onto a problem of q-deformed oscillators, whose real parameter q measures the frustration. We find the analytic solution of this model by suitably representing the q-oscillator algebra with q-Hermite polynomials. We also present a related matrix model which possesses the same diagrammatic expansion in the planar approximation. Its interaction potential is oscillating at infinity with period , and may lead to interesting metastability phenomena beyond the planar approximation. The spherical model is similarly q-periodic, but does not exhibit such phenomena: actually its low-temperature phase is not glassy and depends smoothly on q.

Area versus length distribution for closed random walks

Using a connection between the q-oscillator algebra and the coefficients of the high temperature expansion of the frustrated Gaussian spin model, we derive an exact formula for the number of closed random walks of given length and area, on a hypercubic lattice, in the limit of infinite number of dimensions. The formula is investigated in detail, and asymptotic behaviour is evaluated. The area distribution in the limit of long loops is computed. As a byproduct, we also obtain an infinite set of new, nontrivial identities.

Arctic Curve of the Free-Fermion Six-Vertex Model in an L-Shaped Domain

We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall boundary conditions, in the case of free-fermion vertex weights. We describe how the recently developed ‘Tangent method’ can be used to determine the form of the arctic curve. The obtained result is in agreement with numerics.

Counting non-planar diagrams: an exact formula

Abstract We present an explicit solution of a simply stated, yet unsolved, combinatorial problem, of interest both in quantum field theory (Feynman diagrams enumeration, beyond the planar approximation) and in statistical mechanics (high temperature loop expansion of some frustrated lattice spin model).