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Hughes B.D. — Random Walks and Random Environments: Random Environments (том 2)
Hughes B.D. — Random Walks and Random Environments: Random Environments (том 2)

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Название: Random Walks and Random Environments: Random Environments (том 2)

Автор: Hughes B.D.


This is the second volume of a two-volume work devoted to probability theory in physical chemistry, and engineering. Rather than dealing explicitly with the idea of an ongoing random walk, with each chaotic step taking place at fixed time intervals, this volume addresses models in which the disorder is frozen in space-random environments. It begins with an introduction to the geometry of random environments, emphasizing Bernoulli percolation models. The scope of the investigation then widens as we ask how structural disorder affects the transport process. The final chapters confront the interplay of two different forms of randomness; spatial randomness frozen into the environment and temporal randomness associated with the choices for next steps made by a random walker. The book ends with a discussion of "the ant in the labyrinth" problems and an extensive bibliography that, along with the rest of the material, will be of value to researchers in physics, mathematics, and chemical engineering.

Язык: en

Рубрика: Математика/Вероятность/Стохастические процессы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1996

Количество страниц: 274

Добавлена в каталог: 04.06.2005

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Предметный указатель
AB percolation      55
Acceptance profile      63
Admissible current distribution      323
Admissible potential distribution      321
Alexander — Orbach conjecture      364 458—465 467
Alexander — Orbach scaling law      459 481
Animals      See Lattice animals
Ant in the labyrinth, above percolation threshold      442—443 481
Ant in the labyrinth, at percolation threshold      443—444
Ant in the labyrinth, below percolation threshold      439—442
Ant in the labyrinth, biased      452
Ant in the labyrinth, defined      437
Ant in the labyrinth, dynamical scaling theories, dominant time scales      450—451 456—457
Ant in the labyrinth, dynamical scaling theories, extensions      451
Ant in the labyrinth, dynamical scaling theories, mean number of distinct sites visited      445—447
Ant in the labyrinth, dynamical scaling theories, mean-square displacement      447—448
Ant in the labyrinth, dynamical scaling theories, other applications      448—450
Ant in the labyrinth, rigorous results at $p_c$, Bethe lattice      466—467
Ant in the labyrinth, rigorous results at $p_c$, square lattice      467—468
Ant in the labyrinth, rigorous results for $p>p_c$, Bethe lattice      482
Ant in the labyrinth, rigorous results for $p>p_c$, hypercubic lattice      481
Ant in the labyrinth, Teles' scaling laws      468—481
Ant in the labyrinth, transience above $p_x$ if $d\ge3$      481
Ant in the labyrinth, trapping      494
Ant in the labyrinth, variations      496
Ant, blind      438
Ant, myopic      438
Antipercolation      55
Apparent fractal dimension of the backbone      296
Apparent fractal dimension of the infinite cluster      297—298
Arithmetic distribution      404
Backbone probability, bound for conductivity      335
Backbone probability, bounded      94 107
Backbone probability, for Bethe lattice      26—27
Backbone, apparent fractal dimension      296
Backbone, defined      94
Backbone, exponent bounds      107 271
Backbone, few rigorous results      52
Backbone, fractal model      359
Backbone, introduced for Bethe lattice      26
Backbone, node and link model      358
Backbone, node-link-blob model      360
Backbone, picture of      293
Bernoulli percolation models, bond      See Bond percolation
Bernoulli percolation models, etymology      4
Bernoulli percolation models, site      See Site percolation
Bethe lattice, ant in labyrinth for      466 482
Bethe lattice, backbone probability B(p)      26
Bethe lattice, cluster-size distribution $P_n(p)$ exact for $z=3$, derivation      37
Bethe lattice, cluster-size distribution $P_n(p)$ exact for $z>3$, quoted      37
Bethe lattice, conduction problem on      343
Bethe lattice, connectivity function      244
Bethe lattice, defined      22
Bethe lattice, Durrett — Nguyen inequality sharp for      216
Bethe lattice, effectively $d=\infty$      22
Bethe lattice, exponents $\gamma$ and $\gamma'$ equal      202
Bethe lattice, exponents same as periodic lattice above upper critical dimension      50
Bethe lattice, exponents summarized      198
Bethe lattice, free energy      205
Bethe lattice, gelation model      14
Bethe lattice, infinitely many infinite clusters above $p_c$      133
Bethe lattice, mean cluster size $\chi(p)$      35
Bethe lattice, pair connectedness      244
Bethe lattice, percolation probability $P_\infty(p)$      25
Bethe lattice, percolation threshold of      35
Bethe lattice, picture of      22
Bethe lattice, problems with concavity of F(p, h)      221
Bethe lattice, relatives of      27 37
Bethe lattice, should take $\nu=\frac12$      245
Bethe lattice, site-bond percolation on      54
Bethe lattice, susceptibility $\chi_f(p)$      37
Bipartite lattice      56 175 464
Bipartite percolation      56
Blind ant      438
Boltzmann distribution      451
Boltzmann's constant      451
Bond bridge      21
Bond bridge, closed, defined      5
Bond bridge, open, defined      5
Bond percolation, clusters in      5
Bond percolation, defined      5
Bond percolation, equivalent to site percolation on covering lattice      13
Bond percolation, exact thresholds for d=2, bow-tie lattice      158
Bond percolation, exact thresholds for d=2, history      142—143
Bond percolation, exact thresholds for d=2, honeycomb lattice      156
Bond percolation, exact thresholds for d=2, overview of proofs      143
Bond percolation, exact thresholds for d=2, square lattice $p_c\ge1/2$      147
Bond percolation, exact thresholds for d=2, square lattice $p_c\le1/2$      145
Bond percolation, exact thresholds for d=2, summarized      142
Bond percolation, exact thresholds for d=2, triangular lattice      156
Bond percolation, pictures of      5—11
Bond percolation, PostScript code to simulate      10
Bond percolation, wet set, cardinality of      11
Bond percolation, wet set, defined      11
Borel — Cantelli lemma      492
Braess's paradox      324
Branch conductance      344
Bridge bond      21
Buckingham — Gunton scaling law      249
Bus bars      326
Cardy's conjecture on crossing probabilities      261—264
Cauchy — Schwarz inequality      See also Schwarz inequality
Cauchy — Schwarz inequality, stated      206 219
Cauchy — Schwarz inequality, used      215 219 222 225 227 267 346 413
Cayley tree, conduction problem on      343
Cayley tree, defined      22
Chebyshev's inequality, Markov's inequality as precursor      212
Chebyshev's inequality, used in ergodic theory      132
Chemical distance      476
Cluster shape      51
Cluster-size distribution, $P_n(p)$ defined      15
Cluster-size distribution, decay at low density      38—39
Cluster-size distribution, decay in high-density phase      45—46 125—126
Cluster-size distribution, exponential decay in low-density phase      108—111
Cluster-size distribution, for Bethe lattice      35—37
Commute time      474
Conditional distribution      138
Conditional expectation      114
Conditional expected number of pivotal sites      101
Conditional mean cluster size, given cluster size finite      37
Conditional probability density function for bond conductance after renormalization      351
Conditional probability density function for branch conductance on Bethe lattice      346
Conditional probability distribution for ant's cluster size      439
Conditional probability, $\sigma(s, s_0)$      88 98
Conditional probability, definition recalled      88 101
Conditional probability, in definition of incipient infinite cluster      301
Conditional probability, that occupied site belongs to infinite cluster      28 52
Conditional probability, that open bond belongs to infinite cluster      30
Conditional probability, used      107 114 115 117 137
Conductivity exponent, conjectures      359 360 364—369
Conductivity exponent, dynamical scaling theory for      454
Conductivity exponent, Flory-type argument      361
Conductivity exponent, numerical estimates      369—371
Conductivity, effective      326
Conductivity, microscopic      348
Conductivity, specific      326
Conformal field theory      261
Connective constant, bounding critical probability      16—17
Connective constant, defined      2
Connectivity function, at percolation threshold      238—240
Connectivity function, decay of in low-density phase      97
Connectivity function, defined      88
Connectivity function, for Bethe lattice      244
Connectivity function, in high-density phase      242—244
Connectivity function, moments of      233
Connectivity function, scaling theory for      241
Contact process      69
Continuum percolation      68
Continuum resistance dimension      468
Convex function      328
Coordination number, defined      2 22
Correlated percolation      58
Correlation length, critical exponent $\nu$ for      231
Correlation length, defined for percolation      229
Correlation length, definition via moments of pair-connectedness      233
Correlation length, divergence of      230
Correlation length, exponent bounds      231 235—238
Correlation length, in high-density phase      125 243
Correlation length, relevant to ant in labyrinth      454—456
Cover time      476
Covering lattice      12
Critical amplitudes      266 283
critical exponent      2
Critical exponents for percolation, defined, $\beta$, $\gamma$ and $\delta$      198
Critical exponents for percolation, defined, $\Delta, \Delta_k, \Delta', \Delta_k'$      206—207
Critical exponents for percolation, defined, $\gamma'$, $\alpha$ and $\alpha'$      201—204
Critical exponents for percolation, estimates for d=3      288—289
Critical exponents for percolation, mean-field values      199
Critical exponents for percolation, on Bethe lattice $\gamma=\gamma'$      202
Critical exponents for percolation, overview      50
Critical exponents for percolation, rigorous scaling inequalities, Durrett and Nguyen      215—216
Critical exponents for percolation, rigorous scaling inequalities, Newman's      218—221
Critical exponents for percolation, scaling theories, alternative forms      211
Critical exponents for percolation, scaling theories, of Essam and Gwilym      211
Critical exponents for percolation, scaling theories, Stauffer's      208—210
Critical exponents for percolation, “exact” values for d=2      284—288
Critical probability, for high-density phase ($p_H$), defined      15
Critical probability, for high-density phase ($p_H$), lower bound for square lattice bond problem      147—152
Critical probability, for high-density phase ($p_H$), lower bound for triangular lattice site problem      161—162
Critical probability, for low-density phase ($p_L$), defined      33
Critical probability, for low-density phase ($p_L$), upper bound for square lattice bond problem      145—147
Critical probability, for low-density phase ($p_L$), upper bound for triangular lattice site problem      162
Critical probability, for slice, limit of      125 486
Critical probability, sponge-crossing ($p_S$), defined      124
Critical probability, sponge-crossing ($p_S$), equal to $p_L$ and $p_H$      124. See also percolation threshold
Criticality Theorem of groadbent and Hammersley      16
Crossing probability, Cardy's conjecture      261
Crossing probability, rigorous result for square lattice bond percolation at $p=\frac12$      145
Decimation      252
Dense lattice      478
Dhar's formula      444 458 462 495
Diffusion constant for continuum diffusion      451—453 468
Diffusion constant for Temkin's model      397
Diffusion constant for the ant in the labyrinth      442 453 456
Diffusion constant on the infinite cluster      443 452 454 456
Diffusion process, contrast with percolation      4
Dimension, continuum resistance      468
Dimension, diode models      419
Dimension, first-passage time      477
Dimension, for infinite cluster      297
Dimension, fractal, apparent for backbone      296
Dimension, fractal, in scaling laws for ant in labyrinth      477
Dimension, fracton      458
Dimension, harmonic      444 445 459 462 464 467 481 495
Dimension, resistance      469 477
Dimension, spectral      444 445 458 459 462 464 467 481 495
Directed percolation      See percolation theory variations
Dirichlet's Principle, analogue used      429 430
Dirichlet's Principle, stated and proved      321
Dirichlet's Principle, used      325 327 331 336
Discrete Tauberian Theorem, stated      36
Discrete Tauberian Theorem, used      36 205
Distance, chemical      476
Distance, graphical      476
Distance, stepping      86 89 91 97 230 294 476
Distribution, arithmetic      404
Drag coefficient      451
Dual lattices, applications to random resistor networks      328
Dual lattices, defined      143
Dual lattices, implications on percolation thresholds      152—156
Dynamical phase transition      392
Dynamical scaling theory for conductivity      454—455. See also ant in the labyrinth dynamical
Eden model      68
Effective conductance between sites      321
Effective conductivity      326
Effective conductivity, one dimension      319
Effective conductivity, series and parallel bounds      327
Effective medium approximation, cluster improvements      353
Effective medium approximation, estimates of percolation threshold      353
Effective medium approximation, for bond percolation conduction      352
Effective medium approximation, for site percolation conduction      355
Effective resistance between sites      321
Eigenfunction expansion techniques      393
Eigenvalue, multiple-path diffusion      425
Eigenvalue, renormalization transformation      253
Einstein, Albert      318
Ergodic theorem      131
Ergodic theory, applied to Temkin's model      396
Ergodic theory, introduced      127
Ergodic transformation      129
Euler transformation, used for high-density series      32
Euler transformation, used for low-density series      43
Event approximation lemma, stated but not proved      128
Event approximation lemma, used      129
Event, decreasing      91
Event, increasing      See Increasing events
Event, invariant under transformation      128
Exchange matrix      423
Excursion in Temkin's problem      388
Expected number of clusters per lattice site      207
Face of a plane lattice      144 154 156 168
Finite-size scaling      259 266—267 340
First Brillouin zone      245
First-passage time, dimension      477
First-passage time, mean, related to electrical conduction      473—474
First-passage time, mean, related to mean commute time      475
First-passage time, on square lattice incipient infinite cluster      467
Fisher's Lemma      161
Fixed point, attractive      253
Fixed point, stable      253
Fixed point, unstable      253
FKG inequality      See Harris-FKG inequality
Flory-type argument      361
Flow capacity      337
For Bethe lattice cluster size, conditioned on radius      37
Fortuin — Kasteleyn — Swendsen — Wang distribution      279
Fractal dimension      477
Fractal dimension, apparent, of the backbone      296
Fractal dimension, apparent, of the infinite cluster      297
Fractal model of backbone      359
Fractal structure, effective, effect on ant in labyrinth      442
Fractals, prior knowledge assumed      2
Fracton dimension      458
Fractons      458—459
Free energy, application to ant in labyrinth      439
Free energy, continuous differentiability of      204
Free energy, convexity of modified form      204
Free energy, F(p, h) defined      203
Free energy, zero-field E(p)=F(p,0), defined by series      164
Free energy, zero-field E(p)=F(p,0), equals mean number of clusters per lattice site      164
Free energy, zero-field E(p)=F(p,0), for Bethe lattice      205
Fully triangulated lattice, defined      168
Fully triangulated lattice, van den Berg's counter-example      47
Gamma function, defined      xxiv
Gap exponents, defined      206
Gap exponents, known equal in two dimensions      213 251
Gap exponents, should be equal      210
Generalized master equation      410
Glauber model      70
Graphical distance      476
Grimmett, Kesten, and Zhang, lemma of      481
Grimmett, Kesten, and Zhang, theorem of      481
Hammersley's inequality for site-bond, percolation      53
Hammersley, John M.      3
Harmonic dimension      444 445 459 462 464 467 481 495
Harris — Fisher theorem      154
Harris — FKG inequality, stated and proved      90
Harris — FKG inequality, used      90—92 124 151 152 155 159 229 230 232 240 241 484
Harris' lemma, for a quadrant      147
Harris' lemma, for the half-plane      148
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