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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.

Аннотация:

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

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Random resistor problem, one-dimensional      319—320
Random walk in random environment      386—496
Random walk in random environment, one-dimensional discrete-time, recurrent      See Sinai's problem
Random walk in random environment, one-dimensional discrete-time, transient      See Temkin's model
Random walk in random environment, one-dimensional master equation      See randomized master equation
Random walk in random environment, variations      496
Random walk in random scenery      495
Random walk, additional references for Volume 1      505—507
Random walk, continuous-time      386 405. Chapter
Random walk, Polya type      437. See also Volume 1 Chapters
Random walk, self-avoiding      See Self-avoiding walk
Random-random directed walk      420
Randomized master equation, asymmetric one-dimensional      419—421
Randomized master equation, dimension greater than 1      428—430
Randomized master equation, multistate      421—428
Randomized master equation, symmetric one-dimensional, defined      410—411
Randomized master equation, symmetric one-dimensional, disorder slows exploration      412—413
Randomized master equation, symmetric one-dimensional, formalism of Alexander et al.      411—412
Randomized master equation, symmetric one-dimensional, non-universal case      416—419
Randomized master equation, symmetric one-dimensional, scaling arguments      413—414
Randomized master equation, symmetric one-dimensional, various treatments      415—416
Rayleigh's monotonicity law stated, and proved      324
Rayleigh's monotonicity law stated, used      327 328 349 472 473 478 479 493
Rayleigh, Lord      318
Realization of a percolation process      5
Realization of a random environment in Temkin's problem      387
Recurrent random walk in random environment      388
Renormalization, approximate for square lattice      254—258
Renormalization, exact for Sierpinski lattice      260—261
Renormalization, for percolation conduction problem      350
Renormalization, for Temkin's model      405—406
Renormalization, position-space      252
Renormalization, real-space      252
Renormalization, transformation      252
Renormalized effective medium, approximation      356
Replica method      348 415
Resistance      319
Resistance dimension      469 477
Resistive susceptibility      361
Right excursion, defined      388
Right recurrent, defined      388
Right recurrent, necessary and sufficient conditions      388
Russo — Seymour — Welsh theorem, cf. Cardy's conjecture on crossing probabilities      263
Russo — Seymour — Welsh theorem, stated      159
Russo — Seymour — Welsh theorem, used      240 249
Russo's formula, corollary to      101
Russo's formula, special case of      294
Russo's formula, stated and proved      99
Russo's formula, used      102 106 119
Scalar models      317
Scaling relation of Stauffer for $P_n(p)$, implications for exponents      210
Scaling relation of Stauffer for $P_n(p)$, rigorous for Bethe lattice      210
Scaling relation of Stauffer for $P_n(p)$, stated      209
Scaling relations for percolation, heuristic discussion      2Q7—211
Scaling relations for percolation, rigorous results      215—216 248—251
Schwarz inequality, used      206 212 233 413.
Self-avoiding walk      See also Volume 1 Chapter
Self-avoiding walk, additional references for Volume 1      507—510
Self-avoiding walk, bounds for percolation probability $P_\infty(p)$      17
Self-avoiding walk, prior knowledge assumed      1—2
Self-avoiding walk, used      16 21 23 24 40 58 90 93 102 123
Series expansion, high-density for $P_\infty(p)$      27
Series expansion, low-density for $\chi(p)$      40
Series, resistors arranged in      319
Seymour — Welsh theorem      153
Shape of percolation clusters      51
Sierpinski lattice      260
Simon's inequality, stated and proved      94
Simon's inequality, used      98
Simple plane lattice      168
Sinai's problem      406—410
Sinai's theorem, non-rigorous derivation      407
Sinai's theorem, stated      407
Site percolation, clusters in      11
Site percolation, defined      11—12
Site percolation, exact thresholds for d=2, history      160
Site percolation, exact thresholds for d=2, summarized      160
Site percolation, includes bond percolation      13
Site percolation, picture of, backbone      293
Site percolation, picture of, hull      179
Site percolation, picture of, just above $p_c$      292
Site percolation, wet set, cardinality of      11
Site percolation, wet set, defined      11
Site, empty or vacant      11
Site, p-occupied      107 137
Site, present or occupied      11
Site-bond percolation      52 265
Slowly-varying      36 198 205 401
Smooth lattice      479
Solomon's theorem, on first-passage times in Temkin's model      403
Solomon's theorem, on recurrence of Temkin's model      388
Specific conductivity      326
Specific heat exponents in percolation      204
Spectral dimension      444 445 458 459 462 464 467 481 495
Spineless infinite cluster      104
Sponge      123
Sponge-crossing critical probability      124
Square lattice, $\eta\delta_r=2$      239
Square lattice, $\nu=\nu'$      243
Square lattice, $\nu_k=\nu$      235
Square lattice, exact bond threshold      142
Star-triangle relation      69 156 264
Stepping distance, defined      86
Stepping distance, used      89 91 97 230 294 476 480 481
Stokes — Einstein relation      451—457 461 470
Stokes — Einstein relation, misleading use of term      453
Stokes — Einstein relation, standard use of term      452
Strong law of large numbers, applied to Temkin's problem      389 390 396
Strong law of large numbers, one-dimensional resistor networks      319 332
Susceptibility $\chi_f(p)$ in percolation      37 107 201
Susceptibility $\chi_f(p)$ in percolation, exact for Bethe lattice      37
Swendsen — Wang distribution      279
Swiss-cheese model      339
Symmetric difference      128
Tauberian Theorem, discrete      See Discrete Tauberian
Temkin's model, definition and formulation      387—388
Temkin's model, diffusion constant for      397
Temkin's model, excursions and recurrence      388—390
Temkin's model, first-passage times and excursion times      397—405
Temkin's model, mean duration of a left excursion, averaged over environments      397
Temkin's model, mean duration of a left excursion, distribution over environments      399
Temkin's model, possible behaviours      390—392
Temkin's model, renormalization      405—406
Temkin's model, treatment of Derrida and Pomeau      392—397
Temkin's model, variants      406
Theorem for Laplace transform, stated      413
Theorem for Laplace transform, strong      See Tauberian Theorem for Laplace transform
Theorem for Laplace transform, used      413 414 417
Theorem of Aizenman and Newman on divergence of the mean cluster size      102
Theorem of Aizenman and Newman on the decay of connectivity function      97
Theorem of Aizenman and Newman on the decay of the cluster-size distribution      111
Theorem of Burton and Keane      135
Theorem of Grimmett and Marstrand, stated      125
Theorem of Grimmett and Marstrand, used      482 486
Theorem of Grimmett, Kesten, and Zhang      481
Theorem of Hammersley, van den Berg, and Kesten      95
Theorem of Kesten, Kozlov, and Spitzer      404
Theorem of Men'shikov      See Men'shikov's theorem
Theorem of Newman and Schulman      133
Theorem of Nguyen      234
Theorem of Solomon, on first-passage times in Temkin's model      403
Theorem of Solomon, on recurrence of Temkin's model      388
Theorem of van den Berg and Keane      137
Theorem, Bond-site comparison      20
Theorem, infinite cluster uniqueness      135
Theorem, on recurrence for nested lattices      472
Theorem, on slices and quadrants of slices stated      482
Theorem, used      483—484
Thomson's Principle, stated and proved      323
Thomson's Principle, used      325 327 329 338
Trace operator      275
Transformation, ergodic      129
Transformation, measure-preserving      128
Transition matrix      423
Tree conductance      343
Tree graph      108
Trial current distribution      323
Trial potential distribution      321
Triangle condition, holds for large enough dimension      200
Triangle condition, implications for scaling theory      212
Triangle condition, introduced      200
Triangle condition, significance of      245
Triangle inequality, for real numbers      129 130
Triangle inequality, for stepping distance, stated      89
Triangle inequality, for stepping distance, used      91 98
Triangular lattice, admits Seymour — Welsh Theorem      153
Triangular lattice, exact bond threshold      142
Triangular lattice, exponents same as for honeycomb      200
Triangular lattice, site threshold      160
Triple point      136
Turnstile models      419
Uniqueness of infinite cluster, implications of      126
Uniqueness of infinite cluster, proved      132—136
Uniqueness principle, proof set as exercise      325
Uniqueness principle, used      472 474
Universality      50
Upper critical dimensionality, and hyperscaling      282—283
Upper critical dimensionality, and triangle condition      200—201
Upper critical dimensionality, apparent fractal dimensions above      298
Upper critical dimensionality, believed to be $d_c=6$      50
Upper critical dimensionality, different for directed percolation      291
Upper critical dimensionality, evidence in favour of $d_c=6$      201 272—274
Upper semicontinuous functions      137
Van den Berg-Kesten inequality, stated and proved      93
Van den Berg-Kesten inequality, used      93—95 102 105 110 115 116 244
Variational method      328
Vector models      317
Voronoi algorithm      59 339
Weak convergence      407 418 429
Weak Law of Large Numbers, for one-dimensional resistor networks      319
Wet set, bond percolation, defined      11
Wet set, radius of      94
Wet set, site percolation, defined      11
Wet set, span of      242
Whitney dual of two-dimensional lattice      143
Wigner — Seitz construction      59
“bow-tie” lattice      158 175 200
“dice” lattice      170
“incipient infinite cluster”      248 249 301—303 444—448 466 467
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