Reinforced Random Walks and Adic Transformations |
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| Authors: | Sarah Bailey Frick Karl Petersen |
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| Affiliation: | (1) Movement Science Group, School of Life Sciences, Oxford Brookes University, Gipsy Lane, Headington, Oxford, OX3 0BP, UK;(2) Department of Neurology and Section of Neuroradiology, Medical University Graz, Graz, Austria;(3) Department of Clinical Neurology, University of Oxford, Oxford, UK;(4) School of Life Sciences, Oxford Brookes University, Oxford, UK;(5) Oxford Centre for Enablement, Oxford, UK;(6) Department of Mathematical Sciences, School of Technology, Oxford Brookes University, Oxford, UK;(7) FMRIB, University of Oxford, Oxford, UK;(8) GlaxoSmithKline, Imperial College London, London, UK |
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| Abstract: | To a given finite graph we associate three kinds of adic, or Bratteli–Vershik, systems: stationary, symbol-count, and reinforced. We give conditions for the natural walk measure to be adic-invariant and identify the ergodic adic-invariant measures for some classes of examples. If the walk measure is adic-invariant, we relate its ergodic decomposition to the vector of limiting edge traversal frequencies. For some particular nonsimple reinforcement schemes, we calculate the density function of the edge traversal frequencies explicitly. |
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