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| Crisanti A., Paladin G., Vulpiani A. — Products of random matrices in statistical physics |
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| Предметный указатель |
Adiabatic approximation 87
Affine transformation 128
Affine transformation contractive 128
Anderson model 12 89
Anderson model, two-dimensional 106
Annealed, average 75
Annealed, free energy see "Free energy"
Arnold — Beltrami — Childress (ABC) map 125
Asymptotic limits for PRM see "Furstenberg theorem" "Oseledec
Baker transformation 51—52
Barnsley — Elton theorem 129
Bernoullian probability distribution 14
Borland conjecture 91
Cauchy probability distribution 95
Central limit theorem 22
Central limit theorem for i.i.d. matrices in  23
Central limit theorem for i.i.d. matrices with positive elements 22
Central limit theorem for i.i.d. symplectic matrices 23
Central limit theorem for random numbers 17
Circular theorem 5
Cohen — Newman theorem 29—30
Collage theorem 129
Conductivity 93 see
Conductivity, maximum entropy approach to fluctuations of 110
Correlation function 75
Correlation length, most probable 78
Density of states integrated 91—92 see
Differentiable dynamical systems, continuous time 43
Differentiable dynamical systems, discrete time 43
Dimension, fractal 132 148
Dimension, generalized fractal 126 149
Dimension, information 149 150
Dimension, topological 148
Directed random polymers 32
Discrete Laplace operator in one dimension 89
Discrete Laplace operator in two dimesions 107
Disordered, harmonic chains 8
Disordered, Ising chain 11
Ecosystems 6
Effective Lyapunov exponent see "Maximum Lyapunov exponent"
Eigenvalues see "Statistics of eigenvalues"
Eigenvectors, Lyapunov see "Lyapunov basis"
Eigenvectors, stability see "stability basis"
Electrostatics, two-dimensional classi cal 136
Entropy, function for effective Lyapunov exponent 25
Entropy, function for multifractal objects 149
Entropy, information 113
Entropy, Kolmogorov — Sinai or metric 55 146
Entropy, Renyi 148
Entropy, Shannon 145
Entropy, topological 52 146
Extended states 88
Extended states in light transmission 118
Extended states in random 1d potentials 104—106
Fermi — Pasta — Ulam (FPU) model 56
Fluctuations of exponential growth rate 23
Fluctuations of Lyapunov exponent in canonical ensemble 38
Fluctuations of magnetic field in kinematic dynamo see "Generalized Lyapunov exponents for kinematic dynamo"
Fractal dimension see "Dimension"
Free energy fluctuations 71—72
Free energy fluctuations of random coupling 1d Ising model 74
Free energy, annealed 71
Free energy, of 1d Ising model 60
Free energy, of random 1d Ising model 12 61
Free energy, of random 2d Ising model 84
Free energy, quenched 70 71
Furstenberg theorem 19
Gaussian, measure in matrix space 135 138
Gaussian, orthonormal ensemble (GOE) 4 135
Gaussian, unitary ensemble (GUE) 4
Generalized correlation function 77 78
Generalized correlation function, Monte Carlo calculation of 80
Generalized correlation lengths 77 78
Generalized correlation lengths for random 1d Ising model 80
Generalized correlation lengths for random d > 1 Ising model 85
Generalized correlation lengths for random field 1d Ising model 79
Generalized localization lengths 100—101
Generalized localization lengths for random potentials with power law distribution 104
Generalized localization lengths, weak disorder expansion for 1d Anderson model 102
Generalized Lyapunov exponents 22—23 101
Generalized Lyapunov exponents for i.i.d. gaussian matrices 31
Generalized Lyapunov exponents for kinematic dynamo 124
Generalized Lyapunov exponents in canonical ensemble 39
Generalized Lyapunov exponents in microcanonical ensemble 38
Generalized semicircular theorem 5
Gram — Schmidt orthonormalization method 27 142
Hamiltonian flow 56
Hausdorff distance 129
Henon map 51
Herbert — Jones — Thouless formula for 1d localization 92
Herbert — Jones — Thouless formula for quasi-1d localization 108
Hilbert space 121
Incidence angle 119
Incidence angle, average critical 120
Incidence angle, critical 120
Independent electrons approximation 87
Independent RM approximation for hamiltonian flows 56—57
Independent RM approximation for kinematic dynamo 126
Independent RM approximation for symplectic maps 46
Induction equation 123
Ising model 59
Ising model, random 1d 11
Ising model, random 2d Ising model 82
Ising model, random coupling 1d 61
Ising model, random field 1d 61
Iterated function system (IFS) 129
Iterated function system, average contractive 129 130
Kinematic dynamo 123
Lagrangian, chaos 124
Lagrangian, motion 124
Landauer formula for 1d localization 93
Landauer formula for quasi-1d localization 108
Large number law for i.i.d. matrices in 20
Large number law for i.i.d. matrices with positive elements 20
Large number law for random numbers 17
Legendre tranformation 25 150
| Localization length 12 90
Localization length fluctuations 101
Localization length fluctuations, weak disorder expansion for 1d Anderson model 102
Localization length for quasi-1d localization 108
Localization of eigenstates in random harmonic chains 12
Localization of wave functions in correlated random potential 153
Localization of wave functions in random potential 12
Localized states 88
Lognormal probability distribution 18 24
Low temperature expansion for random field 1d Ising model 63—67
Lozi map 52
Lyapunov basis 21
Lyapunov exponents (CLE) 21
Lyapunov exponents for i.i.d. gaussian matrices 28
Lyapunov exponents, numerical calculation of 27 138
Lyapunov spectrum 21
Magnetic diffusivity 123
Magnetic dynamos see "Kinematic dynamo"
Markov chain 144
Markov chain, ergodic 144
Maximum entropy ansatz 113 see maximum
Maximum Lyapunov exponent 10 19
Maximum Lyapunov exponent for i.i.d. gaussian matrices 31
Maximum Lyapunov exponent for large sparse i.i.d. matrices 31
Maximum Lyapunov exponent, effective 25
Maximum Lyapunov exponent, numerical calculation of 26
Mean return time in Poincare section 45
Metal-insulator transition 109
Microcanonical, ensemble for PRM 36
Microcanonical, method 37
Mott — Twose conjecture 89
Multifractal 126 149
Multifractal measure 125
Nearest-neighbours tight-binding model 89 see
Numerical methods for the computation of Lyapunov exponents see "Lyapunov exponents"
Optical index 118
Oseledec theorem 20
Oseledec theorem in disordered systems 71
Oseledec theorem in dynamical systems 44
Partition function 11
Partition function for 1d Ising model 60
Partition function for random 2d Ising model 83
Penetration length 119 123
Perturbative expansion for 1d localization 96—97
Perturbative expansion for products of symplectic matrices 48—50
Perturbative expansion for random field 1d Ising model 62—64 67
Pesin relation 55 147
Poincare section 142—143
Poincare section, map 143
Poisson equation 136
Polymers see "Directed random polymers"
Probability measure 125 133 149
Product of commutative diagonalizable ma trices 13
Product of random numbers 13
Quantum Ohm law 94
Quenched free energy see "Free energy"
Random energy model 32
Reflection coefficient 94 119
Replica method for 1d localization 98
Replica method for disorderd systems 71
Replica method for product of random matrices 34
Replica method, variational 99
Resonance in optical transmission see "Extended states in optical localization"
Schroedinger equation 88 118
Schroedinger equation, discretized with random potential see "Anderson model"
Self-averaging 71 88
Semicircular theorem 4
Semicircular theorem, generalised semicircular theorem 5
Shannon — McMillan theorem 145
Sierpinsky, carpet 131
Sierpinsky, gasket 130
Stability basis 21
Stability matrix for dynamical systems 10 44
Stability matrix for hamiltonian flows 56
Stability matrix for symplectic maps 46
Stability of ecosystems 7—8
Statistics of eigenvalues, complex random matrices 138
Statistics of eigenvalues, real random matrices 5 135
Statistics of eigenvalues, real random symmetric matrices 4
Superlocalization 104
Symbolic dynamics 146
Symplectic, maps 9 45
Symplectic, matrices 46 107
Tangent map 10
Tangent map for 2d symplectic map 10
Tangent map for Baker transformation 52
Tangent map for Lozi map 52
Tangent vector 10 43
Thermal average 75
Thermodynamic limit for Lyapunov exponents 54
Thermodynamic limit for quasi-1d localization 109
Thermodynamic limit, correction to t.l. in disordered systems 71
Tight-binding approximation see "Anderson model"
Time-evolution operator 43
Transfer matrix for 1d Ising model 60
Transfer matrix for 1d optical media 119 121
Transfer matrix for 2d Anderson model 107
Transfer matrix for Anderson model 90
Transfer matrix for random 1d Ising model 11
Transfer matrix for random 2d Ising model 84
Transfer matrix for random coupling 1d Ising model 61
Transfer matrix for random field 1d Ising model 61
Transfer matrix for random waveguide 122
Transition matrix of Markov process 53 144 152
Transmission coefficient 94 119
Verlet method 56
Virster theorem 22 23
Wave-equation 118
waveguide see "Transfer matrix for random waveguide"
Weak disorder expansion 33
Weak disorder expansion for 1d localization 97
Wiener, paths 127
Wiener, process 127
Zel’dovic antidynamo theorem 127
Zero temperature entropy of random coupling 1d Ising model 69 70
Zero temperature limit for random coupling 1d Ising model 68 69
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