Random walks on the braid group B3 and magnetic translations in hyperbolic geometry
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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