Random walks on the braid group B3 and magnetic translations in hyperbolic geometry |
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| Institution: | 1. Center for Advanced Biomaterials for Health Care @CRIB, Istituto Italiano di Tecnologia, Largo Barsanti e Matteucci 53, 80125 Napoli, Italy;2. Istituto di Ricerche sulla Combustione, Consiglio Nazionale delle Ricerche, P.le Tecchio 80, 80125 Napoli, Italy;3. Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven 5600 MB, The Netherlands;4. Dipartimento di Ingegneria Chimica, dei Materiali e della Produzione Industriale, Universitá di Napoli Federico II, P. le Tecchio 80, 80125 Napoli, Italy |
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| Abstract: | We study random walks on the three-strand braid group B3, and in particular compute the drift, or average topological complexity of a random braid, as well as the probability of trivial entanglement. These results involve the study of magnetic random walks on hyperbolic graphs (hyperbolic Harper–Hofstadter problem), what enables to build a faithful representation of B3 as generalized magnetic translation operators for the problem of a quantum particle on the hyperbolic plane. |
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