Random walks and electric networks by Peter G. Doyle

By Peter G. Doyle

Doyle P.G., Snell J.L. Random walks and electrical networks (New York MAA 1984)(ISBN 0883850249)

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8421053. 8421177. 2 2 We note that E¯ is greater than E, though just barely. Thomson’s Principle. (Thomson [33]). If i is the unit flow from a to b determined by Kirchhoff’s Laws, then the energy dissipation 1 1 2 2 x,y ixy Rxy minimizes the energy dissipation 2 x,y jxy Rxy among all 2 unit flows j from a to b. Proof. Let j be any unit flow from a to b and let dxy = jxy − ixy . Then d is a flow from a to b with da = x dax = 1 − 1 = 0. x,y 2 jxy Rxy = (ixy + dxy )2 Rxy x,y = x,y i2xy Rxy + 2 i2xy Rxy = ixy Rxy dxy + x,y +2 x,y x,y x,y (vx − vy )dxy + d2xy Rxy d2xy Rxy .

14 2 4 3 84 The probability that the reversed sequence d b c a occurs is 5 2 1 1 1 · · · = . 14 5 3 4 84 Thus the two sequences have the same probability of occurring. In general, when a reversible Markov chain is started in equilibrium, probabilities for sequences in the correct order of time are the same as those with time reversed. Thus, from data, we would never be able to tell the direction of time. If P is any reversible ergodic chain, then P is the transition matrix for a random walk on an electric network; we have only to define Cxy = wx Pxy .

Then d is a flow from a to b with da = x dax = 1 − 1 = 0. x,y 2 jxy Rxy = (ixy + dxy )2 Rxy x,y = x,y i2xy Rxy + 2 i2xy Rxy = ixy Rxy dxy + x,y +2 x,y x,y x,y (vx − vy )dxy + d2xy Rxy d2xy Rxy . x,y Setting w = v and j = d in the conservation of energy result above shows that the middle term is 4(va − vb )da = 0. Thus 2 jxy Rxy = x,y i2xy Rxy + x,y x,y d2xy Rxy ≥ i2xy Rxy . x,y This completes the proof. 11 The following is the so-called “dual form” of Thomson’s Principle. Let u be any function on the points of the graph G of a circuit such that ua = 1 and ub = 0.

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