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Table of contents
1 CPL and the qKZ equation
1.1 Completely Packed Loops
1.1.1 Bond percolation
1.1.2 Connectivity
1.1.3 Transfer Matrix and Integrability
1.1.4 The Hamiltonian and the Temperley{Lieb algebra
1.1.5 A stochastic process in the link patterns
1.2 The XXZ Spin Chain Model
1.2.1 The quantum algebra Uq(su(2))
1.2.2 Translation between XXZ Spin Chain model and CPL
1.3 The quantum Knizhnik{Zamolodchikov equation
1.3.1 The operator Si
1.4 Finding a solution
1.4.1 Building a solution
1.4.2 A vector space
1.5 Contour integral formul
1.5.1 The action of the qKZ equation
1.5.2 Space
1.5.3 Link Patterns
1.6 A third basis
1.6.1 A new integral formula
1.6.2 Basis transformation
1.6.3 The homogeneous limit
1.7 An example
2 ASM and 6-Vertex model
2.1 Alternating Sign Matrices
2.2 6-Vertex Model
2.2.1 Square Ice Model and a path model
2.2.2 Partition Function
2.2.3 The case q3 = 1
2.2.4 The homogeneous limit
2.3 Fully Packed Loops
2.3.1 The Razumov{Stroganov{Cantini{Sportiello theorem
2.3.2 Some symmetry classes
3 TSSCPP
3.1 Plane Partitions
3.1.1 Totally Symmetric Self complementary Plane Partitions
3.2 Non-Intersecting Lattice Paths
3.2.1 The LGV formula
3.3 TSSCPP as NILP
3.4 Descending Plane Partitions
4 ASM and TSSCPP
4.1 Generating functions
4.1.1 NILP
4.1.2 ASM
4.2 The conjecture
4.3 Integral formul
4.3.1 ASM counting as an integral formula
4.3.2 NILP counting as an integral formula
4.3.3 Equality between integral formul
4.4 The original conjecture
5 CPL and PP
5.1 Some notation
5.2 Study of entries of the type ()p
5.2.1 a-Basis
5.2.2 Reduction to size r
5.2.3 Expansion for high p
5.2.4 Sum rule
5.2.5 A NILP formula
5.2.6 Punctured{TSSCPP
5.2.7 Limit shape
5.3 Study of entries of the type (p
5.3.1 Basis transformation
5.3.2 NILP
5.3.3 Punctured{TSSCPP
5.3.4 Limit shape
5.4 Further questions
5.4.1 Zuber’s conjectures
5.4.2 Limit shape
6 On the polynomials ()p : some conjectures
6.1 Notes on the FPL case
6.2 The conjectures
6.2.1 Combinatorics
6.2.2 The case = 1
6.2.3 Generic
6.3 Contour integral formula for G
6.3.1 Dual paths
6.3.2 The sum of G
6.3.3 Computing G()n
6.4 The rst root
6.4.1 The proof
6.5 The subleading term of the polynomials
6.5.1 First proof
6.5.2 Second proof
6.5.3 Application to hook length products
6.6 The leading term of (; p)
6.7 Further questions
6.7.1 Solving the conjectures
6.7.2 Combinatorial reciprocity
6.7.3 Consequences of the conjectures
A A vector space
B Proof of Lemma 2.27
C Anti-symmetrization formul
C.1 Proof of the identity (5.20)
C.1.1 The general case
C.1.2 Integral version
C.1.3 Homogeneous Limit
C.2 Proof of the identity (6.15)
D Examples of ()p
E Equivalence of two denitions
F Resume en francais
F.1 Introduction
F.2 Boucles denses et qKZ
F.2.1 Boucles denses
F.2.2 Connectivite
F.2.3 La matrice R et l’integrabilite
F.2.4 Le cas homogene
F.2.5 La cha^ne de spins XXZ
F.2.6 L’equation de Knizhnik{Zamolodchikov quantique
F.2.7 Des integrales de contour
F.2.8 La limite homogene
F.3 Matrices a Signes Alternants et le modele a 6 vertex
F.3.1 Matrices a Signes Alternants
F.3.2 Modele a 6 vertex
F.3.3 La fonction de partition
F.3.4 Boucles compactes
F.4 Partitions planes
F.4.1 Partitions planes totalement symetriques et auto-complementaires
F.4.2 Chemins values non intersectants
F.5 ASM et TSSCPP
F.5.1 Une conjecture ranee
F.5.2 Esquisse de la preuve
F.6 Boucles denses et partitions planes
F.6.1 Polynomialite
F.6.2 La somme des ()p
F.6.3 La somme des (p
F.7 Des polyn^omes ()p : nouvelles conjectures
F.7.1 Denitions
F.7.2 Les conjectures
F.7.3 Questions ouvertes
G Resumo em portugu^es
G.1 Introduc~ao
G.2 Lacetes densos e qKZ
G.2.1 Lacetes densos
G.2.2 Conectividade
G.2.3 A matriz R e a integrabilidade
G.2.4 Caso homogeneo
G.2.5 A cadeia de spins XXZ
G.2.6 A equac~ao Knizhnik{Zamolodchikov qu^antica
G.2.7 Integrais de contorno
G.2.8 O limite homogeneo
G.3 Matrizes de Sinal Alternante e o modelo 6-vertices
G.3.1 Matrizes de Sinal Alternante
G.3.2 Modelo 6-vertices
G.3.3 A func~ao de partic~ao
G.3.4 Lacetes compactos
G.4 Partic~oes planas
G.4.1 Partic~oes planas totalmente simetricas e auto-complementares .
G.4.2 Caminhos ponderados n~ao intersectantes
G.5 ASM e TSSCPP
G.5.1 Uma conjectura renada
G.5.2 Esquisso da prova
G.6 Lacetes densos e partic~oes planas
G.6.1 Polinomialidade
G.6.2 A soma dos ()p
G.6.3 A soma dos (p
G.7 Polinomios ()p : novas conjecturas
G.7.1 Algumas denic~oes
G.7.2 As conjecturas
G.7.3 Quest~oes abertas
References
