Random Walks and Electric Networks

Random Walks and Electric Networks /

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Probability theory, like much of mathematics, is indebted to physics as a source of problems and intuition for solving these problems. Unfortunately, the level of abstraction of current mathematics often makes it difficult for anyone but an expert to appreciate this fact. Random Walks and Electric N...

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Bibliographic Details
Online Access: Full text (MCPHS users only)
Other Authors: Doyle, Peter G., Snell, J. Laurie
Format: Electronic eBook
Language:English
Published: Cambridge : Cambridge University Press, 2012
Series:Carus.
Subjects:
Random walks (Mathematics)
Electric network topology.
Local Note:ProQuest Ebook Central
Table of Contents:
  • Front Cover
  • Random Walks and Electric Networks
  • Copyright Page
  • Preface
  • Contents
  • Part I: Random Walks on Finite Networks
  • Chapter 1. Random Walks in One Dimension
  • 1.1. A random walk along Madison Avenue
  • 1.2. The same problem as a penny matching game
  • 1.3. The probability of winning: basic properties
  • 1.4. An electric network problem: the same problem?
  • 1.5. Harmonic functions in one dimension; the Uniqueness Principle
  • 1.6. The solution as a fair game (martingale)
  • Chapter 2. Random Walks in Two Dimensions
  • 2.1. An example
  • 2.2. Harmonic functions in two dimensions2.3. The Monte Carlo solution
  • 2.4. The original Dirichlet problem; the method of relaxations
  • 2.5. Solution by solving linear equations
  • 2.6. Solution by the method of Markov chains
  • Chapter 3. Random Walks on More General Networks
  • 3.1. General resistor networks and reversible Markov chains
  • 3.2. Voltages for general networks; probabilistic interpretation
  • 3.3. Probabilistic interpretation of current
  • 3.4. Effective resistance and the escape probability
  • 3.5. Currents minimize energy dissipation
  • Chapter 4. Rayleigh's Monotonicity Law4.1. Rayleigh's Monotonicity Law
  • 4.2. A probabilistic explanation of the Monotonicity Law
  • 4.3. A Markov chain proof of the Monotonicity Law
  • Part II: Random Walks on Infinite Networks
  • Chapter 5. Pólya's Recurrence Problem
  • 5.1. Random walks on lattices
  • 5.2. The question of recurrence
  • 5.3. Polya's original question
  • 5.4. Polya's Theorem: recurrence in the plane, transience in space
  • 5.5. The escape probability as a limit of escape probabilities for finite graphs
  • 5.6. Electrical formulation of the type problem5.7. One dimension is easy, but what about higher dimensions?
  • 5.8. Getting around the lack of rotational symmetry of the lattice
  • 5.9. Rayleigh: shorting shows recurrence in the plane, cutting shows transience in space
  • Chapter 6. Rayleigh's Short-Cut Method
  • 6.1. Shorting and cutting
  • 6.2. The Shorting Law and the Cutting Law; Rayleigh's idea
  • 6.3. The plane is easy
  • 6.4. Space: searching for a residual network
  • 6.5. Trees are easy to analyze
  • 6.6. The full binary tree is too big
  • 6.7. NT3: a three-dimensional tree6.8. NT3 has finite resistance
  • 6.9. But does NT3 fit in the three-dimensional lattice?
  • 6.10. What we have done; what we will do
  • Chapter 7. The Classical Proofs of Pólya's Theorem
  • 7.1. Recurrence is equivalent to an infinite expected number of returns
  • 7.2. Simple random walk in one dimension
  • 7.3. Simple random walk in two dimensions
  • 7.4. Simple random walk in three dimensions
  • 7.5. The probability of return in three dimensions: exact calculations

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