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Remerciements iii

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Contents

Abstract i

Remerciements iii

Introduction 1

1 Large N vector models 19

1.1 Definition of the models . . . . 20

1.2 Feynman graphs . . . . 21

1.3 Large N expansion . . . . 23

1.4 Leading sector: bubble graphs . . . . 26

1.5 Discussion . . . . 29

2 Large N matrix models 31 2.1 Definition of the models . . . . 32

2.2 Feynman graphs . . . . 34

2.2.1 Stranded representation . . . . 34

2.2.2 Colored representation . . . . 36

2.3 Large N expansion . . . . 38

2.4 Leading sector: planar graphs . . . . 40

2.5 Discussion . . . . 42

2.5.1 Case of complex matrices . . . . 42

2.5.2 Case of reduced symmetry . . . . 43

3 Large N tensor models 45 3.1 Definition of the models . . . . 48

3.2 Feynman graphs . . . . 50

3.2.1 Stranded representation . . . . 50

3.2.2 Colored representation . . . . 51

3.3 Results on d-bubbles . . . . 53

3.3.1 Jackets and degree . . . . 53

3.3.2 3-bubble subgraphs and index . . . . 56

3.3.3 Addition formulas . . . . 58

3.4 Large N expansion (1): BGR scaling . . . . 59

3.4.1 Large N expansion . . . . 59

3.4.2 Leading sector: melonic graphs . . . . 62

3.4.3 Discussion . . . . 68

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3.5 Large N expansion (2): enhanced scaling . . . . 68

3.5.1 Large N expansion . . . . 69

3.5.2 Leading sector: generalized melonic graphs . . . . 72

3.5.3 Optimal scalings, MST interactions and mirror melons . . . . 74

3.5.4 Case of the complete interaction . . . . 75

3.5.5 Discussion and outlook . . . . 81

4 Large N and large D matrix-tensor models 85 4.1 Definition of the models . . . . 88

4.2 Feynman graphs . . . . 89

4.2.1 Stranded representation . . . . 90

4.2.2 Colored representation . . . . 90

4.3 Large N and large D expansions . . . . 92

4.3.1 Large N and large D scalings . . . . 92

4.3.2 Case of the enhanced scaling . . . . 94

4.4 Application: quantum models and SYK physics . . . . 99

4.5 Discussion and outlook . . . . 102

5 More on the New Large D Limit of Matrix Models 107 5.1 Introduction and summary . . . . 109

5.2 Definition of the models . . . . 110

5.3 Vertices and graphs . . . . 111

5.4 The large N and large D limits . . . . 112

5.4.1 Counting the power of N . . . . 112

5.4.2 Counting the power of D . . . . 114

5.4.3 Form of the expansions and leading order graphs . . . . 115

5.4.4 On cases with reduced symmetry . . . . 116

5.5 On correlation functions . . . . 119

5.5.1 General remarks . . . . 119

5.5.2 Connected 2n-point correlation functions . . . . 120

5.6 Model building . . . . 122

5.6.1 Unstable bosonic models . . . . 122

5.6.2 Stable bosonic models . . . . 125

5.6.3 Supersymmetric models . . . . 126

6 Melonic Turbulence 133 6.1 Introduction . . . . 135

6.2 Resonant systems . . . . 138

6.2.1 The model . . . . 138

6.2.2 Tree expansion of non-linear flows in one dimension . . . . 141

6.2.3 Tree expansion for the case under study . . . . 144

6.2.4 Averaged Sobolev norms . . . . 148

6.2.5 Large order heuristic analysis . . . . 151

6.3 Explicit computations at order t

2

. . . . 153

6.3.1 Amplitudes at order 2 . . . . 153

6.3.2 Sobolev norms at order 2 . . . . 157

6.4 Melonic dominance . . . . 159

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6.4.1 Melonic graphs . . . . 160

6.4.2 Stranded representation . . . . 162

6.4.3 Existence of the 1/N expansion . . . . 163

6.4.4 Melonic dominance . . . . 171

6.4.5 Exponential bound for melonic graphs and analyticity of the mel- onic Sobolev norms . . . . 175

Appendices 187 A Graphs embedded on surfaces 189 A.1 Abstract graphs . . . . 189

A.1.1 General definitions . . . . 189

A.1.2 Edge-colored graphs and bubbles . . . . 191

A.2 Embedded graphs . . . . 193

A.2.1 Cellularly embedded graphs . . . . 194

A.2.2 Ribbon graphs . . . . 194

A.2.3 Signed rotation system . . . . 195

A.2.4 Genus of embedded graphs . . . . 197

B Classification of PCGMs 199 B.1 Properties of the complete graph and its edge-coloring . . . . 199

B.1.1 The MST condition . . . . 199

B.1.2 Distinguishing edges and vertices . . . . 200

B.2 Action and index . . . . 202

B.3 The classification theorem . . . . 202

B.3.1 Useful tools . . . . 202

B.3.2 Results on faces . . . . 203

B.3.3 The PCGM with two bubbles . . . . 206

B.3.4 The most general PCGMs . . . . 208

C A few remarks on MST interactions 213 C.1 The complete bipartite interaction . . . . 213

C.2 Building MSTs from MSTs . . . . 214

vii

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