Random walks and electrical networks

Imagine a random walk on a graph G starting at x and ending at either a or z. Then the probability p(x) that the random walk ends at a is a harmonic function. Interpreting G as an electrical network, the voltage u(x) induced by a unit voltage applied between a and z is a harmonic function, too. Thus the two functions p(x) and u(x), being equal at a and z, coincide. Consequently, many electrical quantities, like current, effective conductance, or effective resistance, have interesting probabilistic interpretations. Hitting times, for instance, can be computed by calculating effective resistances. In the present diploma thesis this is demonstrated for the Ehrenfest urn model. The above approach is generalized for infinite networks by replacing z with "infinity". One of the first questions arising in this context is, whether a random walk on an infinite network is recurrent or transient. Based on the above connections between random walks and electrical networks efficient recurrence criteria have been found. Two of the most famous criteria are the flow criterion by Yamasaki/Lyons and the Nash-Williams criterion. In this diploma thesis the most important probabilistic interpretations of electrical quantities are presented with unified notation and proof methods. The author gives many instructive examples and several important recurrence criteria.