Six-vertex, Loop and Tiling Models: Integrability and Combinatorics
نویسنده
چکیده
This is a review (including some background material) of the author’s work and related activity on certain exactly solvable statistical models in two dimensions, including the six-vertex model, loop models and lozenge tilings. Applications to enumerative combinatorics and to algebraic geometry are described.
منابع مشابه
The Rotor Model and Combinatorics
The XXZ Heisenberg spin chain and the related six-vertex model stand as central pillars in the study of exactly solved models in statistical mechanics. It has been known for many years that, with appropriate boundary conditions, their groundstate energy is trivial at the particular anisotropy value ∆ = −1/2. Only recently has it been realised that the corresponding groundstate wavefunction poss...
متن کاملDomino tilings and the six-vertex model at its free fermion point
At the free-fermion point, the six-vertex model with domain wall boundary conditions (DWBC) can be related to the Aztec diamond, a domino tiling problem. We study the mapping on the level of complete statistics for general domains and boundary conditions. This is obtained by associating to both models a set of non-intersecting lines in the Lindström-Gessel-Viennot (LGV) scheme. One of the conse...
متن کاملThe free energies of six-vertex models and the n-equivalence relation
The free energies of six-vertex models on a general domain D with various boundary conditions are investigated with the use of the n-equivalence relation, which help classify the thermodynamic limit properties. It is derived that the free energy of the six-vertex model on the rectangle is unique in the limit height,width → , . It is derived that the free energies of the model on the domain D ar...
متن کاملThermodynamic limit of the Six - Vertex Model with Domain Wall Boundary Conditions
We address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model. Here we compare the bulk free energy for periodic and domain wall boundary conditions. Using a determinant representation for the partition function with domain wall boundary conditions, we derive Toda differential equations and solve them asymptotically in order to extract the bul...
متن کاملValence-partitioned genus polynomials and their application to generalized dipoles
Calculations of genus polynomials are given for three kinds of dipoles: with no loops; with a loop at one vertex; or with a loop at both vertices. We include a very concise, elementary derivation of the genus polynomial of a loopless dipole. To describe the general effect on the face-count and genus polynomials of the operation of adding a loop at a vertex, we introduce imbedding types that are...
متن کامل