An electric network for non-reversible Markov chains

Marton Balazs, Aron Folly

Research output: Contribution to journalArticle (Academic Journal)peer-review

3 Citations (Scopus)
276 Downloads (Pure)

Abstract

We give an analogy between non-reversible Markov chains and electric networks much in the flavour of the classical reversible results originating from Kakutani, and later Kemeny-Snell-Knapp and Kelly. Non-reversibility 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 non-reversible result of Gaudilliere-Landim 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 languageEnglish
Pages (from-to)657-682
Number of pages26
JournalAmerican Mathematical Monthly
Volume123
Issue number7
DOIs
Publication statusPublished - 19 Aug 2016

Keywords

  • Non-reversible Markov chains
  • Electric networks
  • Effective resistance
  • Absorption probability
  • Commute time

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