Quantum Probability and Spectral Analysis of Graphs (eBook)

Quantum Probability and Orthogonal Polynomials.- Adjacency Matrix.- Distance-Regular Graph.- Homogeneous Tree.- Hamming Graph.- Johnson Graph.- Regular Graph.- Comb Graph and Star Graph.- Symmetric Group and Young Diagram.- Limit Shape of Young Diagrams.- Central Limit Theorem for the Plancherel Measure of the Symmetric Group.- Deformation of Kerov's Central Limit Theorem.- References.- Index.

Foreword 5
Preface 10
Contents 14
1 Quantum Probability and Orthogonal Polynomials 18
1.1 Algebraic Probability Spaces 18
1.2 Representations 23
1.3 Interacting Fock Probability Spaces 28
1.4 The Moment Problem and Orthogonal Polynomials 31
1.5 Quantum Decomposition 40
1.6 The Accardi–Bozejko Formula 45
1.7 Fermion, Free and Boson Fock Spaces 53
1.8 Theory of Finite Jacobi Matrices 59
1.9 Stieltjes Transform and Continued Fractions 68
Exercises 76
Notes 79
2 Adjacency Matrices 81
2.1 Notions in Graph Theory 81
2.2 Adjacency Matrices and Adjacency Algebras 83
2.3 Vacuum and Deformed Vacuum States 86
2.4 Quantum Decomposition of an Adjacency Matrix 91
Exercises 96
Notes 99
3 Distance-Regular Graphs 100
3.1 Definition and Some Properties 100
3.2 Spectral Distributions in the Vacuum States 103
3.3 Finite Distance-Regular Graphs 106
3.4 Asymptotic Spectral Distributions 109
3.5 Coherent States in General 115
Exercises 116
Notes 118
4 Homogeneous Trees 119
4.1 Kesten Distribution 119
4.2 Asymptotic Spectral Distributions in the Vacuum State ( Free CLT) 123
4.3 The Haagerup State 124
4.4 Free Poisson Distribution 132
4.5 Spidernets and Free Meixner Law 134
4.6 Markov Product of Positive Definite Kernels 139
Exercises 142
Notes 143
5 Hamming Graphs 145
5.1 Definition and Some Properties 145
5.2 Asymptotic Spectral Distributions in the Vacuum State 148
5.3 Poisson Distribution 150
5.4 Asymptotic Spectral Distributions in the Deformed Vacuum States 154
Exercises 159
Notes 160
6 Johnson Graphs 161
6.1 Definition and Some Properties 161
6.2 Asymptotic Spectral Distributions in the Vacuum State 166
6.3 Exponential Distribution and Laguerre Polynomials 168
6.4 Geometric Distribution and Meixner Polynomials 170
6.5 Asymptotic Spectral Distributions in the Deformed Vacuum States 173
6.6 Odd Graphs 180
Exercises 185
Notes 187
7 Regular Graphs 188
7.1 Integer Lattices 188
7.2 Growing Regular Graphs 190
7.3 Quantum Central Limit Theorems 195
7.4 Deformed Vacuum States 202
7.5 Examples and Remarks 206
Exercises 214
Notes 215
8 Comb Graphs and Star Graphs 217
8.1 Notions of Independence 217
8.2 Singleton Condition and Central Limit Theorems 222
8.3 Integer Lattices and Homogeneous Trees: Revisited 228
8.4 Monotone Trees and Monotone Central Limit Theorem 231
8.5 Comb Product 241
8.6 Comb Lattices 245
8.7 Star Product 250
Exercises 256
Notes 257
9 The Symmetric Group and Young Diagrams 260
9.1 Young Diagrams 260
9.2 Irreducible Representations of the Symmetric Group 264
9.3 The Jucys–Murphy Element 268
9.4 Analytic Description of a Young Diagram 270
9.5 A Basic Trace Formula 274
9.6 Plancherel Measures 278
Exercises 280
Notes 281
10 The Limit Shape of Young Diagrams 282
10.1 Continuous Diagrams 282
10.2 The Limit Shape of Young Diagrams 286
10.3 The Modified Young Graph 288
10.4 Moments of the Jucys–Murphy Element 291
10.5 The Limit Shape as a Weak Law of Large Numbers 294
10.6 More on Moments of the Jucys–Murphy Element 296
10.7 The Limit Shape as a Strong Law of Large Numbers 304
Exercises 306
Notes 306
11 Central Limit Theorem for the Plancherel Measures of the Symmetric Groups 308
11.1 Kerov’s Central Limit Theorem and Fluctuation of Young Diagrams 308
11.2 Use of Quantum Decomposition 310
11.3 Quantum Central Limit Theorem for Adjacency Matrices 312
11.4 Proof of QCLT for Adjacency Matrices 317
11.5 Polynomial Functions on Young Diagrams 321
11.6 Kerov’s Polynomials 324
11.7 Other Extensions of Kerov’s Central Limit Theorem 325
11.8 More Refinements of Fluctuation 328
Exercises 330
12 Deformation of Kerov’s Central Limit Theorem 332
12.1 Jack Symmetric Functions 332
12.2 Jack Graphs 336
12.3 Deformed Young Diagrams 338
12.4 Jack Measures 341
12.5 Deformed Adjacency Matrices 345
12.6 Central Limit Theorem for the Jack Measures 351
12.7 The Metropolis Algorithm and Hanlon’s Theorem 356
Exercises 360
Notes 360
References 362
Index 373