Random walks and random environments: 2 Random environments
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| Format: | Buch |
| Sprache: | English |
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Oxford
Clarendon Press
1996
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| Schriftenreihe: | Oxford science publications
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| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xxiv, 526 Seiten Illustrationen |
| ISBN: | 0198537891 |
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MARC
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| 100 | 1 | |a Hughes, Barry D. |0 (DE-588)171914430 |4 aut | |
| 245 | 1 | 0 | |a Random walks and random environments |n 2 |p Random environments |c Barry D. Hughes |
| 264 | 1 | |a Oxford |b Clarendon Press |c 1996 | |
| 300 | |a xxiv, 526 Seiten |b Illustrationen | ||
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Datensatz im Suchindex
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CONTENTS
Notes on the Text xix
Introduction to Volume 2 1
1 An Introduction to Percolation Theory 4
1.1 The Problem of Broadbent and Hammersley 4
1.1.1 The Origins of the Percolation Model 4
1.1.2 The (Bernoulli) Bond Percolation Model 5
1.1.3 The (Bernoulli) Site Percolation Model 11
1.1.4 An Overview of this Chapter 13
1.1.5 Remarks on the Literature 13
1.2 The Percolation Probability 15
1.2.1 The High density Phase 15
1.2.2 Comparison of Site and Bond Percolation
Probabilities 20
1.2.3 The Percolation Probability for the Bethe
Lattice 22
1.2.4 High density Series 27
1.2.5 An Example of High density Series
Analysis 30
1.3 The Cluster size Distribution 33
1.3.1 The Low density Phase 33
1.3.2 The One dimensional Percolation
Problem 34
1.3.3 Cluster size Distribution for the
Bethe Lattice 35
1.3.4 The Decay with n of Pn{p) at Low
Density 38
1.3.5 Low density Series 40
1.3.6 The Decay with n of Pn(p) at High
Density 45
1.4 Fundamental Problems in Percolation Theory 46
1.4.1 The Threshold Problem 46
1.4.2 Counter examples 47
1.4.3 Locating Thresholds 49
1.4.4 Behaviour Near The Percolation
Threshold 50
1.4.5 Structural Issues 51
xii CONTENTS
1.5 Variations on the Percolation Model 52
1.5.1 Site bond Percolation 52
1.5.2 Polychromatic Percolation and AB
Percolation 54
1.5.3 Topologically Disordered Lattices 58
1.5.4 Directed Percolation 59
1.5.5 An Exact Solution of Domany and Kinzel 60
1.5.6 Invasion Percolation 63
1.5.7 Long range Percolation Models 64
1.5.8 Partially Oriented Percolation Models 66
1.5.9 Continuum Percolation Models 68
1.5.10 Other Variations on the Percolation
Model 69
References for Chapter 1 71
2 Bernoulli Site Percolation 86
2.1 Introduction 86
2.1.1 What We Shall Prove 86
2.1.2 Some Important Definitions 88
2.2 Increasing Events 89
2.2.1 The Harris FKG Inequality 90
2.2.2 An Application of the Harris FKG
Inequality 91
2.2.3 The van den Berg Kesten Inequality 92
2.2.4 Bounds on the Distribution of the Wet
Set Radius 94
2.2.5 A Theorem of Aizenman and Newman 97
2.3 Russo's Formula and its Implications 99
2.3.1 Pivotal Sites and Russo's Formula 99
2.3.2 Divergence of the Mean Cluster Size 101
2.3.3 Bounds on the Slope of the Percolation
Probability 104
2.4 Decay of the Cluster size Distribution 108
2.4.1 Tree graph Inequalities 108
2.4.2 Decay of Pn{p) 111
2.5 Men'shikov's Theorem 112
2.5.1 The Theorem and an Important Lemma 112
2.5.2 Proof of the Theorem 114
2.5.3 Sponge crossing Probabilities 123
2.5.4 Sequences of Critical Probabilities 125
2.6 Uniqueness of the Infinite Cluster 126
2.6.1 Elements of Ergodic Theory 127
2.6.2 Proof of Uniqueness of the Infinite
Cluster 132
CONTENTS xiii
2.6.3 Continuity of the Percolation Probability 136
References for Chapter 2 138
3 Percolation Thresholds 142
3.1 Some Exact Bond Thresholds 142
3.1.1 Overview 142
3.1.2 Dual Lattices in Two Dimensions 143
3.1.3 An Upper Bound for j l for the Square
Lattice 145
3.1.4 A Lower Bound for pu for the Square
Lattice 147
3.1.5 General Implications of Duality 152
3.1.6 The Honeycomb and Triangular Lattices 156
3.1.7 Other Approaches 158
3.2 Some Exact Site Thresholds 160
3.2.1 Overview 160
3.2.2 The Triangular Lattice Site Problem 161
3.2.3 The Mean Number of Clusters Per
Lattice Site 163
3.2.4 Matching Lattices 165
3.3 Bounds, Conjectures, and Empirical Formulae 168
3.3.1 Bounds and Expansions 168
3.3.2 Conjectured Relations To Random Walk
Return Probabilities 172
3.3.3 Other Conjectures and Empirical
Formulae 173
3.4 Numerical Values of Thresholds 175
3.4.1 History 175
3.4.2 Monte Carlo Simulation Algorithms 177
3.4.3 Selected Estimates for Bernoulli
Percolation Models 181
3.4.4 Selected Estimates for Other Models 184
References for Chapter 3 188
4 Critical Exponents in Percolation Theory 197
4.1 Exponents from the Cluster size Distribution 198
4.1.1 The Principal Exponents /?, 7, and S 198
4.1.2 The Triangle Condition and the Upper
Critical Dimension 200
4.1.3 The Exponents 7', a, and a' 201
4.1.4 The Gap Exponents A, A*, A', A'k 206
4.2 Scaling Relations 207
4.2.1 Introduction 207
4.2.2 Stauffer's Scaling Theory 208
xiv CONTENTS
4.2.3 Some Evidence in Support of the Scaling
Hypothesis . 211
4.2.4 A Rigorous Scaling Inequality 215
4.3 Lattice Animal Expansions 216
4.3.1 Expansion of the Cluster size
Distribution 216
4.3.2 Newman's Inequalities 218
4.3.3 Concavity of a Modified Free Energy 221
4.3.4 The Mean square Cluster Size 224
4.3.5 Continuity of Poo(p) at the Percolation
Threshold 227
4.4 Exponents from the Pair Connectedness 229
4.4.1 The Low density Phase and the
Exponent v 229
4.4.2 Moments of the Pair Connectedness 233
4.4.3 The Pair Connectedness at the Threshold 238
4.4.4 Scaling Theory for the Pair
Connectedness 241
4.4.5 The High density Phase and the
Exponent v' 241
4.4.6 The Bethe Lattice and the High dimension
Limit 244
4.4.7 Exact Scaling Relations in Two
Dimensions 248
4.5 Renormalization and Finite size Scaling 252
4.5.1 Basic Concepts 252
4.5.2 Bond Percolation on the Square Lattice 254
4.5.3 Bond Percolation on the Sierpinski
Lattice 260
4.5.4 Critical Crossing Probabilities Revisited 261
4.6 Further Developments of Finite size Scaling 266
4.6.1 Fluctuations in Finite Systems 266
4.6.2 Hyperscaling Inequalities in the High density
Phase 267
4.6.3 The Upper Critical Dimensionality 272
4.7 The Statistical Mechanical Analogy 274
4.7.1 Percolation and the Potts Model 274
4.7.2 Mean field Theory 280
4.7.3 The Upper Critical Dimension and
Hyperscaling 282
4.8 Numerical Values of Critical Exponents 283
4.8.1 'Exact' Values in Two Dimensions 284
4.8.2 Exponent Estimates in Three Dimensions 288
CONTENTS xv
4.8.3 Critical Exponent Estimates for Directed
Percolation 291
4.9 Geometry at the Percolation Threshold 291
4.9.1 The Backbone and its Exponent 291
4.9.2 The Incipient Infinite Cluster 301
References for Chapter 4 303
5 Transport and Conduction in Random
Environments 316
5.1 Overview of Chapters 5 7 316
5.2 The Random Resistor Problem 319
5.2.1 The Random Resistor Problem in One
Dimension 319
5.2.2 General Resistor Networks 320
5.2.3 Dirichlet's Principle 321
5.2.4 Thomson's Principle 323
5.2.5 Rayleigh's Monotonicity Law 324
5.3 The Conductivity of a Random Resistor Network 325
5.3.1 Definition of the Conductivity of a
Random Lattice 326
5.3.2 Rigorous Bounds on the Effective
Conductivity 327
5.3.3 Duality Arguments 328
5.3.4 Remarks on Non linear Random Resistor
Problems 332
5.4 The Percolation Conduction Problem 333
5.4.1 The Basic Model 333
5.4.2 Exponent Inequalities 335
5.4.3 Variations 338
5.5 Scaling Arguments 340
5.5.1 Finite size Scaling 340
5.5.2 Weak Upper Bounds on t 341
5.5.3 Straley's Scaling Theory 342
5.6 Exact Results for the Bethe Lattice 343
5.6.1 Tree and Branch Conductances 343
5.6.2 Ambiguities in Exponent Definitions 347
5.6.3 Potential Correlations in a Random Bethe
Lattice 348
5.7 Uncontrolled Approximations 350
5.7.1 Position space Renormalization 350
5.7.2 Effective Medium Approximations for the
Bond Problem 352
5.7.3 Effective Medium Approximations for the
Site Problem 355
xvi CONTENTS
5.7.4 Renormalized Effective Medium
Approximation 356
5.8 Structural Speculations 358
5.8.1 The Node and Link Model 358
5.8.2 Fractal Structure 359
5.8.3 Series Expansions 361
5.8.4 Flory Arguments 361
5.9 Other Speculations 364
5.9.1 The Alexander Orbach Conjecture 364
5.9.2 Other Conjectures 366
5.10 Numerical Results 369
References for Chapter 5 372
6 Random Walk in a Random Environment 386
6.1 Temkin's Model 387
6.1.1 A Random Walk with a Position dependent
Bias 387
6.1.2 Excursions and Recurrence 388
6.1.3 Classification of Possible Behaviours 390
6.1.4 The Arguments of Derrida and Pomeau 392
6.2 First passage Times in Temkin's Model 397
6.2.1 The Mean Duration of a Left Excursion 397
6.2.2 Distribution over Environments of Mean
Left Excursion Duration 399
6.2.3 First passage Times to Downwind Sites 401
6.2.4 Renormalization Schemes 405
6.2.5 Variants 406
6.3 Sinai's Problem 406
6.3.1 Sinai's Theorem 407
6.3.2 The Asymptotic Distribution of Xn(u) 408
6.4 Randomized Master Equations 410
6.4.1 A Master Equation with Random
Transition Rates 410
6.4.2 The Formalism of Alexander et al. 411
6.4.3 Disorder Slows Exploration 412
6.4.4 Scaling Arguments 413
6.4.5 Other Treatments 415
6.4.6 Non universal Limiting Behaviour 416
6.5 Some Generalizations 419
6.5.1 Asymmetric Models 419
6.5.2 Multistate Models 421
6.5.3 Remarks on Higher dimensional Systems 428
References for Chapter 6 431
CONTENTS xvii
7 The Ant in the Labyrinth 437
7.1 Introduction and Simple Observations 437
7.1.1 Myopic Ants and Blind Ants 437
7.1.2 A Case Study Below the Percolation
Threshold 439
7.1.3 Above the Percolation Threshold 442
7.1.4 At the Percolation Threshold 443
7.2 Dynamical Scaling Theories 444
7.2.1 The Mean Number of Distinct Sites
Visited 445
7.2.2 The Mean square Displacement 447
7.2.3 Other Applications of Dynamical Scaling 448
7.2.4 Dominant Time Scales 450
7.2.5 Extensions 451
7.3 The Stokes Einstein Relation 451
7.3.1 The Original Form of the Stokes Einstein
Relation 451
7.3.2 The Relation between Diffusion Constant
and Conductivity 453
7.3.3 Dynamical Scaling Analysis of the
Conductivity 454
7.3.4 Time Scales Revisited 456
7.4 The Alexander Orbach Conjecture 458
7.4.1 Practons 458
7.4.2 Conjecture for the Harmonic Dimension 459
7.4.3 Implications of the Alexander Orbach
Conjecture 461
7.4.4 Tests of the Alexander Orbach
Conjecture 462
7.5 Some Rigorous Results of Kesten 466
7.5.1 Branches of a Bethe Lattice 466
7.5.2 Square Lattice Bond Percolation 467
7.6 Teles' Scaling Laws 468
7.6.1 Heuristics 469
7.6.2 Polya Walks and Electric Networks 470
7.6.3 Rigorous Derivation of Scaling Laws 476
7.7 Recurrence and Transience for the Ant in the
Labyrinth 481
7.7.1 The Main Results 481
7.7.2 Some Technical Lemmas on Critical
Probabilities 482
7.7.3 Proof of the Lemma of Grimmett, Kesten,
and Zhang 486
xviii CONTENTS
7.8 Variations 494
7.8.1 Trapping the Ant in the Labyrinth 494
7.8.2 Random Walk in Random Sceneries 495
7.8.3 Other Variations 496
References for Chapter 7 496
Errata for the First Printing of Volume 1 503
Additional References for Volume 1 505
Index 516 |
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| spelling | Hughes, Barry D. (DE-588)171914430 aut Random walks and random environments 2 Random environments Barry D. Hughes Oxford Clarendon Press 1996 xxiv, 526 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Oxford science publications (DE-604)BV010207147 2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007432217&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hughes, Barry D. Random walks and random environments |
| title | Random walks and random environments |
| title_auth | Random walks and random environments |
| title_exact_search | Random walks and random environments |
| title_full | Random walks and random environments 2 Random environments Barry D. Hughes |
| title_fullStr | Random walks and random environments 2 Random environments Barry D. Hughes |
| title_full_unstemmed | Random walks and random environments 2 Random environments Barry D. Hughes |
| title_short | Random walks and random environments |
| title_sort | random walks and random environments random environments |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007432217&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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