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 dimensions -- 2.3. 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 Law -- 4.1. 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 problem -- 5.7. 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" tree -- 6.8. 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 -- 7.6. Simple random walk in two dimensions is the same as two independent one-dimensional random walks -- 7.7. Simple random walk in three dimensions is not the same as three independent random walks -- Chapter 8. Random Walks on More General Infinite Networks -- 8.1. Random walks on infinite networks -- 8.2. The type problem -- 8.3. Comparing two networks -- 8.4. The k-fuzz of a graph -- 8.5. Comparing general graphs with lattice graphs -- 8.6. Solving the type problems by flows-a variant of the cutting method -- 8.7. A proof, using flows, that simple random walk in three dimensions is transient -- 8.8. The end -- References -- Index.