Six-Vertex, Loop and Tiling models: Integrability and Combinatorics

Abstract

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.

habilitation thesis v3: this a version from circa 2010, with lots of typos and small inaccuracies fixed compared to the published version

References in corpus (11)

• Emptiness formation probability in the domain-wall six-vertex model
• Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics
• Totally Symmetric Self-Complementary Plane Partitions and Quantum Knizhnik-Zamolodchikov equation: a conjecture
• Exact solution of the six-vertex model with domain wall boundary condition. Critical line between ferroelectric and disordered phases
• The Influence of Boundary Conditions in the Six-Vertex Model
• Combinatorial point for higher spin loop models
• Punctured plane partitions and the q-deformed Knizhnik--Zamolodchikov and Hirota equations
• Bethe roots and refined enumeration of alternating-sign matrices
• Factorised solutions of Temperley-Lieb $q$KZ equations on a segment
• From Orbital Varieties to Alternating Sign Matrices
• Proof of Razumov-Stroganov conjecture for some infinite families of link patterns

Cited by in corpus (40)

• Stochastic six-vertex model
• Free fermions and tau-functions
• Current Fluctuations of the Stationary ASEP and Six-Vertex Model
• Exact results for Wilson loops in arbitrary representations
• Arctic curves of the six-vertex model on generic domains: the Tangent Method
• Quantum Integrability and Generalised Quantum Schubert Calculus
• On the weighted enumeration of alternating sign matrices and descending plane partitions
• Factorial Schur functions and the Yang-Baxter equation
• Special polynomials related to the supersymmetric eight-vertex model: A summary
• Multiply-refined enumeration of alternating sign matrices
• Koornwinder polynomials and the XXZ spin chain
• New enumeration formulas for alternating sign matrices and square ice partition functions
• The density profile of the six vertex model with domain wall boundary conditions
• Generalized string equations for double Hurwitz numbers
• Static properties of 2D spin-ice as a sixteen-vertex model
• Fully Packed Loops in a triangle: matchings, paths and puzzles
• KP and Toda tau functions in Bethe ansatz
• Interacting double dimer model on the square lattice
• Symmetric functions and wavefunctions of the six-vertex model by Izergin-Korepin analysis
• Pfaffian structures and certain solutions to BKP hierarchies II. Multiple integrals
• Special polynomials related to the supersymmetric eight-vertex model. I. Behaviour at cusps
• Lattice Supersymmetry in the Open XXZ Model: An Algebraic Bethe Ansatz Analysis
• Connectivity patterns in loop percolation I: the rationality phenomenon and constant term identities
• Noncommutative fiber products and lattice models
• Planar maps and random partitions
• A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain
• Type $\hat{\mathrm{C}}$ Brauer loop schemes and loop model with boundaries
• On Schur multiple zeta functions: A combinatoric generalization of multiple zeta functions
• Cylindric Hecke characters and Gromov-Witten invariants via the asymmetric six-vertex model
• Gauge-Free Duality in Pure Square Spin Ice: Topological Currents and Monopoles
• Bijective combinatorial proof of the commutation of transfer matrices in the dense O(1) loop model
• Unitary matrix integrals, symmetric polynomials, and long-range random walks
• On some polynomials enumerating Fully Packed Loop configurations
• Loop Models and $K$-Theory
• Lattice Models, Hamiltonian Operators, and Symmetric Functions
• Comments on Slavnov products, Temperley-Lieb open spin chains, and KP tau functions
• Inhomogeneous spin $q$-Whittaker polynomials
• Schur Polynomials and the Yang-Baxter equation
• On some polynomials enumerating Fully Packed Loops configurations, evaluation at negative values
• McMahon's Formula via Free Fermions