Abstract
We give an analogy between nonreversible Markov chains and electric networks much in the flavour of the classical reversible results originating from Kakutani, and later KemenySnellKnapp and Kelly. Nonreversibility is made possible by a voltage multiplier  a 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.
Original language  English 

Pages (fromto)  657682 
Number of pages  26 
Journal  American Mathematical Monthly 
Volume  123 
Issue number  7 
DOIs  
Publication status  Published  19 Aug 2016 
Keywords
 Nonreversible Markov chains
 Electric networks
 Effective resistance
 Absorption probability
 Commute time
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Profiles

Dr Marton Balazs
 School of Mathematics  Reader in Probability
 Probability, Analysis and Dynamics
 Probability
Person: Academic , Member