Potential Theory on Infinite Networks
The aim of the book is to give a unified approach to new developments in discrete potential theory and infinite network theory. The author confines himself to the finite energy case, but this does not result in loss of complexity. On the contrary, the fun
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen
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Paolo M. Soardi
Potential Theory on Infinite Networks
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Paolo M. Soardi Dipartimento di Matematica Universita di Milano Via Saldini, 50 20133 Milano, Italy E-Mail: [email protected]
Mathematics Subject Classification (1991): 05C, 31C, 43A, 52C, 60B, 60J, 94C
ISBN 3-540-58448-X Springer-Verlag Berlin Heidelberg New York CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany Typesetting: Camera ready by author SPIN: 10130158 46/3140-543210 - Printed on acid-free paper
CONTENTS
Introduction
vii
Chapter I. Kirchhoff's laws O. Examples of infinite electrical networks 1. Graphs and networks 2. Chains and cochains 3. Kirchhoff's equations 4. The associated Markov chain 5. Harmonic functions
1 1 3 8 12 14 18
Chapter II. Finite networks 1. Existence and uniqueness of currents 2. The effective resistance 3. Three basic principles
22 22 25 28
Chapter III. Currents and potentials with finite energy 1. Finite energy and Dirichlet sums 2. Existence of currents with finite energy 3. The minimal current 4. Transient networks 5. Recurrent networks 6. Dirichlet's and Rayleigh's principles 7. Decomposition and approximation of functions in D 8. Extremal length 9. Limits of Dirichlet functions along paths
32 32 36 39 43 46 51 58 63 67
Chapter IV. Uniqueness and related topics 1. Spaces of cycles 2. Bounded automorphisms 3. Cartesian products 4. Nonuniqueness and ends 5. The strong isoperimetric inequality 6. Graphs embedded in the hyperbolic disk 7. Nonuniqueness and hyperbolic graphs 8. Moderate growth and Foster's averaging formula
72 72 74 77 79 81 89 93 96
Chapter V. Some examples and computations 1. Transience of infinite grids 2. Potentials in 3. Ungrounded cascades 4. Grounded cascades. Generalized networks
zn
100 100 104 109 113
VI
5. The strong isoperimetric inequality for trees 6. Capacity of the boundary of a tree 7. Edge graphs of tilings of the plane
119
121 125
Chapter VI. Royden's compactiftcation The algebra BD Properties of Royden's compactification Boundary points The harmonic measure Spaces HD of finite dimension Compactification of subnetworks Chapter VII. Rough isometries 1. Rough isometries in metric spaces 2. Some lemmas 3. Recurrence, class OHn and rough isometr
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