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By Russell Lyons, Yuval Peres

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Sometimes, voltages are called potentials or potential differences. Voltages v are then established at every vertex and current i runs through the edges. These functions are observed in experiments to satisfy Ohm’s Law and Kirchhoff’s Law. Physically, Ohm’s law, which is usually stated as v = ir in engineering, is an empirical statement about linear response to voltage differences—certain components obey this law over a wide range of voltage differences. Kirchhoff’s node law expresses the fact that charge does not build up at a node (current being the passage rate of charge per unit time).

5. Suppose that each edge in the following network has equal conductance. What is P[a → z]? We may assume that the edge conductances are all 1, since the probability is not affected by a change in scale of the conductances. Following the transformations indicated in the figure, we obtain C (a ↔ z) = 7/12, so that P[a → z] = † DRAFT 7/12 C (a ↔ z) 7 = = . π(a) 3 36 A conductor c is an edge with conductance c. c ⃝1997–2014 by Russell Lyons and Yuval Peres. Commercial reproduction prohibited. Version of 27 February 2014.

Commercial reproduction prohibited. Version of 27 February 2014. DRAFT 26 Chap. 1. A harmonic function on a 40 × 40 square grid with 4 specified values where it is not harmonic. * But now we call the weights of the edges conductances; their reciprocals are called resistances. ) We denote c(x, y)−1 by r(x, y). The reason for this and more new terminology is that not only does it match physics, but the physics can aid our intuition. We will carefully define everything in pure mathematical terms, but also give a little of the physical background.

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Probability on Trees and Networks by Russell Lyons, Yuval Peres


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