Balázs, Márton; Folly, Áron
(2016):
An Electric Network for Nonreversible Markov Chains.
In: American Mathematical Monthly, Vol. 123, No. 7: pp. 657682

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Abstract
We give an analogy between nonreversible Markov chains and electric networks much in the flavor of the classical reversible results originating from Kakutani and later KemenySnellKnapp and Kelly. Nonreversibility is made possible by a voltage multipliera new electronic component. We prove that absorption probabilities, escape probabilities, expected number of jumps over edges, and commute times can be computed from electrical properties of the network as in the classical case. The central quantity is still the effective resistance, which we do have in our networks despite the fact that individual parts cannot be replaced by a simple resistor. We rewrite a recent nonreversible result of GaudilliereLandim about the Dirichlet and Thomson principles into the electrical language. We also give a few tools that can help in reducing and solving the network. The subtlety of our network is, however, that the classical Rayleigh monotonicity is lost.