 | Duits D. The Hermitian two matrix model with an even quartic potential / M.Duits, A.B.J.Kuijlaars, M.Y.Mo. - Providence: American Mathematical Society, 2012. - v, 105 p. - (Memoirs of the American Mathematical Society; vol.217, N 1022). - Bibliogr.: p.99-102. - Ind.: p.103-105. - ISSN 0065-9266; ISBN 978-0-8218-6928-4
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Chapter 1 Introduction and Statement of Results ................ 1
1.1 Hermitian two matrix model ................................. 1
1.2 Background ................................................. 1
1.3 Vector equilibrium problem ................................. 3
1.4 Solution of vector equilibrium problem ..................... 6
1.5 Classification into cases .................................. 8
1.6 Limiting mean eigenvalue distribution ...................... 9
1.7 About the proof of Theorem 1.4 ............................ 10
1.8 Singular cases ............................................ 12
Chapter 2 Preliminaries and the Proof of Lemma 1.2 ............ 15
2.1 Saddle point equation and functions Sj .................... 15
2.2 Values at the saddles and functions θj .................... 16
2.3 Large z asymptotics ....................................... 17
2.4 Two special integrals ..................................... 18
2.5 Proof of Lemma 1.2 ........................................ 18
Chapter 3 Proof of Theorem 1.1 ................................ 21
3.1 Results from potential theory ............................. 21
3.2 Equilibrium problem for ν3 ................................ 23
3.3 Equilibrium problem for ν1 ................................ 25
3.4 Equilibrium problem for ν2 ................................ 26
3.5 Uniqueness of the minimizer ............................... 27
3.6 Existence of the minimizer ................................ 27
3.7 Proof of Theorem 1.1 ...................................... 29
Chapter 4 A Riemann Surface ................................... 31
4.1 The g-functions ........................................... 31
4.2 Riemann surface  and ξ-functions ........................ 34
4.3 Properties of the ξ functions ............................. 37
4.4 The λ functions ........................................... 38
Chapter 5 Pearcey Integrals and the First Transformation ...... 41
5.1 Definitions ............................................... 41
5.2 Large z asymptotics ....................................... 43
5.3 First transformation: Y → X ............................... 45
5.4 RH problem for X .......................................... 48
Chapter 6 Second Transformation Y → U ......................... 53
6.1 Definition of second transformation ....................... 53
6.2 Asymptotic behavior of U .................................. 53
6.3 Jump matrices for U ....................................... 55
6.4 RH problem for U .......................................... 58
Chapter 7 Opening of Lenses ................................... 61
7.1 Third transformation U → T ................................ 61
7.2 RH problem for T .......................................... 65
7.3 Jump matrices for T ....................................... 66
7.4 Fourth transformation T → S ............................... 68
7.5 RH problem for S .......................................... 70
7.6 Behavior of jumps as n → ∞ ................................ 72
Chapter 8 Global Parametrix ................................... 75
8.1 Statement of RH problem ................................... 75
8.2 Riemann surface as an M-curve ............................. 76
8.3 Canonical homology basis .................................. 77
8.4 Meromorphic differentials ................................. 79
8.5 Definition and properties of functions uj ................. 81
8.6 Definition and properties of functions uj ................. 84
8.7 The first row of M ........................................ 85
8.8 The other rows of M ....................................... 86
Chapter 9 Local Parametrices and Final Transformation ......... 89
9.1 Local parametrices ........................................ 89
9.2 Final transformation ...................................... 94
9.3 Proof of Theorem 1.4 ...................................... 95
Bibliography ................................................... 99
Index ......................................................... 103
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