Random walks on the BMW monoid: an algebraic approach
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We consider Metropolis-based systematic scan algorithms for generating Birman–Murakami–Wenzl (BMW) monoid basis elements of the BMW algebra. As the BMW monoid consists of tangle diagrams, these scanning strategies can be rephrased as random walks on links and tangles. We also consider the Brauer algebra and use Metropolis-based scans to generate Brauer diagrams, giving rise to random walks on perfect matchings. Taking an algebraic perspective, we translate these walks into left multiplication operators in the BMW algebra and so give an algebraic interpretation of the Metropolis algorithm in this setting.
KeywordsMetropolis algorithm Systematic scans Random walks Representation theory Semisimple algebras
The author would like to especially thank Arun Ram for his interest in the work and many helpful conversations, as well as Daniel Rockmore for his guidance and support. The author would also like to thank two anonymous referees for the many helpful suggestions that greatly improved the paper.
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