Random walks and electric networks:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
[Washington, DC]
Mathematical Assoc. of America
1984
|
| Ausgabe: | 1. print. |
| Schriftenreihe: | The Carus mathematical monographs
22 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 159 S. graph. Darst. |
| ISBN: | 0883850249 |
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| 100 | 1 | |a Doyle, Peter G. |e Verfasser |4 aut | |
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| 264 | 1 | |a [Washington, DC] |b Mathematical Assoc. of America |c 1984 | |
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| 650 | 4 | |a Random walks (Mathematics) | |
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Datensatz im Suchindex
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| adam_text |
CONTENTS
Page
Preface vii
Part I. Random Walks on Finite Networks 1
Chapter 1. Random Walks in One Dimension 3
1.1. A random walk along Madison Avenue 3
1.2. The same problem as a penny matching game 4
1.3. The probability of winning: basic properties 4
1.4. An electric network problem: the same problem? 5
1.5. Harmonic functions in one dimension; the Uniqueness
Principle 8
1.6. The solution as a fair game (martingale) 11
Chapter 2. Random Walks in Two Dimensions 15
2.1. An example 15
2.2. Harmonic functions in two dimensions 16
2.3. The Monte Carlo solution 18
2.4. The original Dirichlet problem; the method of relaxations 22
2.5. Solution by solving linear equations 27
2.6. Solution by the method of Markov chains 31
Chapter 3. Random Walks on More General Networks . 39
3.1. General resistor networks and reversible Markov chains . 39
3.2. Voltages for general networks; probabilistic interpretation 45
3.3. Probabilistic interpretation of current 49
3.4. Effective resistance and the escape probability 53
3.5. Currents minimize energy dissipation 60
Chapter 4. Rayleigh's Monotonicity Law 65
4.1. Rayleigh's Monotonicity Law 65
4.2. A probabilistic explanation of the Monotonicity Law 69
4.3. A Markov chain proof of the Monotonicity Law 78
xi
xii CONTENTS
Part II. Random Walks on Infinite Networks 83
Chapter 5. Polya's Recurrence Problem 85
5.1. Random walks on lattices 85
5.2. The question of recurrence 86
5.3. P61ya's original question 88
5.4. Polya's Theorem: recurrence in the plane, transience in
space 88
5.5. The escape probability as a limit of escape probabilities
for finite graphs 89
5.6. Electrical formulation of the type problem 91
5.7. One dimension is easy, but what about higher
dimensions? 93
5.8. Getting around the lack of rotational symmetry of the
lattice 95
5.9. Rayleigh: shorting shows recurrence in the plane, cutting
shows transience in space 96
Chapter 6. Rayleigh's Short Cut Method 99
6.1. Shorting and cutting 99
6.2. The Shorting Law and the Cutting Law; Rayleigh's idea . 100
6.3. The plane is easy 101
6.4. Space: searching for a residual network 102
6.5. Trees are easy to analyze 103
6.6. The full binary tree is too big 106
6.7. JVT3: a "three dimensional" tree 108
6.8. NT3 has finite resistance 110
6.9. But does NT3 fit in the three dimensional lattice? Ill
6.10. What we have done; what we will do 117
Chapter 7. The Classical Proofs of Polya's Theorem 119
7.1. Recurrence is equivalent to an infinite expected number of
returns 119
7.2. Simple random walk in one dimension 120
7.3. Simple random walk in two dimensions 122
7.4. Simple random walk in three dimensions 123
7.5. The probability of return in three dimensions: exact
calculations 125
7.6. Simple random walk in two dimensions is the same as two
independent one dimensional random walks 127
7.7. Simple random walk in three dimensions is not the same
as three independent random walks 128
contents xiii
Chapter 8. Random Walks on More General Infinite
Networks 131
8.1. Random walks on infinite networks 131
8.2. The type problem 132
8.3. Comparing two networks 132
8.4. The fc fuzz of a graph 134
8.5. Comparing general graphs with lattice graphs 136
8.6. Solving the type problems by flows—a variant of the
cutting method 142
8.7. A proof, using flows, that simple random walk in three
dimensions is transient 144
8.8. The end 147
References 151
Index 155 |
| adam_txt |
CONTENTS
Page
Preface vii
Part I. Random Walks on Finite Networks 1
Chapter 1. Random Walks in One Dimension 3
1.1. A random walk along Madison Avenue 3
1.2. The same problem as a penny matching game 4
1.3. The probability of winning: basic properties 4
1.4. An electric network problem: the same problem? 5
1.5. Harmonic functions in one dimension; the Uniqueness
Principle 8
1.6. The solution as a fair game (martingale) 11
Chapter 2. Random Walks in Two Dimensions 15
2.1. An example 15
2.2. Harmonic functions in two dimensions 16
2.3. The Monte Carlo solution 18
2.4. The original Dirichlet problem; the method of relaxations 22
2.5. Solution by solving linear equations 27
2.6. Solution by the method of Markov chains 31
Chapter 3. Random Walks on More General Networks . 39
3.1. General resistor networks and reversible Markov chains . 39
3.2. Voltages for general networks; probabilistic interpretation 45
3.3. Probabilistic interpretation of current 49
3.4. Effective resistance and the escape probability 53
3.5. Currents minimize energy dissipation 60
Chapter 4. Rayleigh's Monotonicity Law 65
4.1. Rayleigh's Monotonicity Law 65
4.2. A probabilistic explanation of the Monotonicity Law 69
4.3. A Markov chain proof of the Monotonicity Law 78
xi
xii CONTENTS
Part II. Random Walks on Infinite Networks 83
Chapter 5. Polya's Recurrence Problem 85
5.1. Random walks on lattices 85
5.2. The question of recurrence 86
5.3. P61ya's original question 88
5.4. Polya's Theorem: recurrence in the plane, transience in
space 88
5.5. The escape probability as a limit of escape probabilities
for finite graphs 89
5.6. Electrical formulation of the type problem 91
5.7. One dimension is easy, but what about higher
dimensions? 93
5.8. Getting around the lack of rotational symmetry of the
lattice 95
5.9. Rayleigh: shorting shows recurrence in the plane, cutting
shows transience in space 96
Chapter 6. Rayleigh's Short Cut Method 99
6.1. Shorting and cutting 99
6.2. The Shorting Law and the Cutting Law; Rayleigh's idea . 100
6.3. The plane is easy 101
6.4. Space: searching for a residual network 102
6.5. Trees are easy to analyze 103
6.6. The full binary tree is too big 106
6.7. JVT3: a "three dimensional" tree 108
6.8. NT3 has finite resistance 110
6.9. But does NT3 fit in the three dimensional lattice? Ill
6.10. What we have done; what we will do 117
Chapter 7. The Classical Proofs of Polya's Theorem 119
7.1. Recurrence is equivalent to an infinite expected number of
returns 119
7.2. Simple random walk in one dimension 120
7.3. Simple random walk in two dimensions 122
7.4. Simple random walk in three dimensions 123
7.5. The probability of return in three dimensions: exact
calculations 125
7.6. Simple random walk in two dimensions is the same as two
independent one dimensional random walks 127
7.7. Simple random walk in three dimensions is not the same
as three independent random walks 128
contents xiii
Chapter 8. Random Walks on More General Infinite
Networks 131
8.1. Random walks on infinite networks 131
8.2. The type problem 132
8.3. Comparing two networks 132
8.4. The fc fuzz of a graph 134
8.5. Comparing general graphs with lattice graphs 136
8.6. Solving the type problems by flows—a variant of the
cutting method 142
8.7. A proof, using flows, that simple random walk in three
dimensions is transient 144
8.8. The end 147
References 151
Index 155 |
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| author | Doyle, Peter G. Snell, James Laurie 1925-2011 |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 1. print. |
| era | Geschichte 1963-1980 gnd Geschichte 1950-1962 gnd Geschichte 1947-1949 gnd |
| era_facet | Geschichte 1963-1980 Geschichte 1950-1962 Geschichte 1947-1949 |
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| id | DE-604.BV022095400 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:14:55Z |
| indexdate | 2025-01-10T19:04:49Z |
| institution | BVB |
| isbn | 0883850249 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015310251 |
| oclc_num | 11764766 |
| open_access_boolean | |
| owner | DE-706 |
| owner_facet | DE-706 |
| physical | XIII, 159 S. graph. Darst. |
| publishDate | 1984 |
| publishDateSearch | 1984 |
| publishDateSort | 1984 |
| publisher | Mathematical Assoc. of America |
| record_format | marc |
| series | The Carus mathematical monographs |
| series2 | The Carus mathematical monographs |
| spelling | Doyle, Peter G. Verfasser aut Random walks and electric networks Peter G. Doyle ; J. Laurie Snell 1. print. [Washington, DC] Mathematical Assoc. of America 1984 XIII, 159 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier The Carus mathematical monographs 22 Geschichte 1963-1980 gnd rswk-swf Geschichte 1950-1962 gnd rswk-swf Geschichte 1947-1949 gnd rswk-swf Marches aléatoires (Mathématiques) PROCESSOS DE MARKOV larpcal Réseaux électriques (Circuits) - Topologie Waarschijnlijkheid (statistiek) gtt Electric network topology Random walks (Mathematics) Einfache Gruppe (DE-588)4123107-7 gnd rswk-swf Irrfahrtsproblem (DE-588)4162442-7 gnd rswk-swf Geschichte (DE-588)4020517-4 gnd rswk-swf Fehlerkorrekturcode (DE-588)4124917-3 gnd rswk-swf Elektrisches Netzwerk (DE-588)4014214-0 gnd rswk-swf Kugelpackung (DE-588)4165929-6 gnd rswk-swf Zufällige Folge (DE-588)4191092-8 gnd rswk-swf Netzwerk (DE-588)4171529-9 gnd rswk-swf Netzwerk (DE-588)4171529-9 s DE-604 Zufällige Folge (DE-588)4191092-8 s Elektrisches Netzwerk (DE-588)4014214-0 s Irrfahrtsproblem (DE-588)4162442-7 s 1\p DE-604 Fehlerkorrekturcode (DE-588)4124917-3 s Geschichte 1947-1949 z Einfache Gruppe (DE-588)4123107-7 s Geschichte (DE-588)4020517-4 s Kugelpackung (DE-588)4165929-6 s Geschichte 1950-1962 z Geschichte 1963-1980 z Snell, James Laurie 1925-2011 Verfasser (DE-588)108435423 aut The Carus mathematical monographs 22 (DE-604)BV001887236 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015310251&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Doyle, Peter G. Snell, James Laurie 1925-2011 Random walks and electric networks The Carus mathematical monographs Marches aléatoires (Mathématiques) PROCESSOS DE MARKOV larpcal Réseaux électriques (Circuits) - Topologie Waarschijnlijkheid (statistiek) gtt Electric network topology Random walks (Mathematics) Einfache Gruppe (DE-588)4123107-7 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd Geschichte (DE-588)4020517-4 gnd Fehlerkorrekturcode (DE-588)4124917-3 gnd Elektrisches Netzwerk (DE-588)4014214-0 gnd Kugelpackung (DE-588)4165929-6 gnd Zufällige Folge (DE-588)4191092-8 gnd Netzwerk (DE-588)4171529-9 gnd |
| subject_GND | (DE-588)4123107-7 (DE-588)4162442-7 (DE-588)4020517-4 (DE-588)4124917-3 (DE-588)4014214-0 (DE-588)4165929-6 (DE-588)4191092-8 (DE-588)4171529-9 |
| title | Random walks and electric networks |
| title_auth | Random walks and electric networks |
| title_exact_search | Random walks and electric networks |
| title_exact_search_txtP | Random walks and electric networks |
| title_full | Random walks and electric networks Peter G. Doyle ; J. Laurie Snell |
| title_fullStr | Random walks and electric networks Peter G. Doyle ; J. Laurie Snell |
| title_full_unstemmed | Random walks and electric networks Peter G. Doyle ; J. Laurie Snell |
| title_short | Random walks and electric networks |
| title_sort | random walks and electric networks |
| topic | Marches aléatoires (Mathématiques) PROCESSOS DE MARKOV larpcal Réseaux électriques (Circuits) - Topologie Waarschijnlijkheid (statistiek) gtt Electric network topology Random walks (Mathematics) Einfache Gruppe (DE-588)4123107-7 gnd Irrfahrtsproblem (DE-588)4162442-7 gnd Geschichte (DE-588)4020517-4 gnd Fehlerkorrekturcode (DE-588)4124917-3 gnd Elektrisches Netzwerk (DE-588)4014214-0 gnd Kugelpackung (DE-588)4165929-6 gnd Zufällige Folge (DE-588)4191092-8 gnd Netzwerk (DE-588)4171529-9 gnd |
| topic_facet | Marches aléatoires (Mathématiques) PROCESSOS DE MARKOV Réseaux électriques (Circuits) - Topologie Waarschijnlijkheid (statistiek) Electric network topology Random walks (Mathematics) Einfache Gruppe Irrfahrtsproblem Geschichte Fehlerkorrekturcode Elektrisches Netzwerk Kugelpackung Zufällige Folge Netzwerk |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015310251&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001887236 |
| work_keys_str_mv | AT doylepeterg randomwalksandelectricnetworks AT snelljameslaurie randomwalksandelectricnetworks |