Introduction .................................................. vii
Giapter 1. Unitary Matrix Ensembles ............................. 1
1.1 Unitary ensemble with real analytic interaction ............ 1
1.2 Ensemble of eigenvalues .................................... 3
1.3 Recurrence equations and discrete string equations for
orthogonal polynomials ..................................... 9
1.4 Deformation equations for the recurrence coefficients ..... 13
1.5 Differential equations and Lax pair for the Ψ-functions ... 16
Chapter 2. The Riemann-Hilbert Problem for Orthogonal
Polynomials .................................................... 19
2.1 The Cauchy transform and its properties ................... 19
2.2 The Riemann-Hilbert problem ............................... 20
2.3 Distribution of eigenvalues and equilibrium measure ....... 22
2.4 The Deift - Zhou steepest descent method .................. 27
2.5 Solution of the RHP for XN(z) ............................. 45
2.6 Asymptotics of the recurrence coefficients ................ 47
2.7 Universality in the random matrix model ................... 50
Chapter 3. Discrete Orthogonal Polynomials on an Infinite
Lattice ........................................................ 55
3.1 The discrete log gas ensemble ............................. 55
3.2 Interpolation problem ..................................... 56
3.3 Equilibrium measure ....................................... 57
3.4 The g-function ............................................ 61
3.5 Reduction of IP to RHP .................................... 62
3.6 First transformation of the RHP ........................... 65
3.7 Second transformation of the RHP .......................... 66
3.8 Model RHP ................................................. 67
3.9 Parametrix at band-void edge points ....................... 68
3.10 Parametrix at the band-saturated region end points ........ 70
3.11 The third and final transformation of the RHP ............ 74
3.12 Asymptotics of recurrence coefficients ................... 75
3.13 Universality in the discrete log gas ensemble ............ 76
Chapter 4. Introduction to the Six-Vertex Model ................ 81
4.1 Definition of the model ................................... 81
4.2 Height function and reduction of parameters ............... 82
4.3 Mappings of the six-vertex model onto other ensembles ..... 84
4.4 Exact solution of the six-vertex model for a finite n ..... 88
Chapter 5. The Izergin-Korepin Formula ......................... 93
5.1 The Yang-Baxter equation .................................. 96
5.2 A proof of Proposition 5.1.1 - ........................... 100
5.3 The recursion equation for Zn ............................ 101
5.4 The inhomogeneous model on the free fermion line ......... 103
5.5 The homogeneous limit .................................... 105
Chapter 6. Disordered Phase ................................... 109
6.1 Main results ............................................. 109
6.2 Rescaling of the weight .................................. 113
6.3 Equilibrium measure ...................................... 114
6.4 Riemann - Hilbert analysis ............................... 128
6.5 Estimates on the jumps for Xn ............................ 133
6.6 Evaluation of X1 ......................................... 135
6.7 Proof of Proposition 6.1.2 ............................... 139
6.8 The constant term ........................................ 139
Chapter 7. Antiferroelectric Phase ............................ 143
7.1 Introduction ............................................. 143
7.2 Jacobi theta functions: Definitions and properties ....... 144
7.3 Main result: Asymptotics of the partition function ....... 148
7.4 Equilibrium measure ...................................... 149
7.5 Riemann - Hilbert analysis ............................... 160
7.6 Evaluation of X1 ......................................... 173
7.7 The constant term ........................................ 192
Chapter 8. Ferroelectric Phase ................................ 197
8.1 Introduction and formulation of the main results ......... 197
8.2 Meixner polynomials ...................................... 198
8.3 Two interpolation problems ............................... 200
8.4 Evaluation of the ratio hk/hkQ ........................... 201
8.5 Evaluation of the constant factor ........................ 204
8.6 Ground state configuration ............................... 206
Chapter 9. Between the Phases ................................. 209
9.1 The critical line between the ferroelectric and
disordered phases ........................................ 209
9.2 The critical line between the antiferroelectric and
disordered phases ........................................ 212
9.3 The order of the phase transitions ....................... 214
Bibliography .................................................. 221
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