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Percolation on Hyperbolic Groups

Speaker: Mahan Mj, Tata Institute of Fundamental Research, Mumbai

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Abstract: We study first passage percolation (FPP) in a Gromov-hyperbolic group G with boundary equipped with the Patterson-Sullivan measure. We associate an .i.d. collection of random passage times to each edge of a  Cayley graph of G, and investigate classical questions about the asymptotics of first passage time as well as the geometry of geodesics in the FPP metric. Under suitable conditions on the passage time distribution, we show that the ‘velocity’ exists in almost every direction, and is almost surely constant by ergodicity of the G-action on the boundary. For every point on the boundary, we also show almost sure coalescence of any two geodesic rays directed towards the point. Finally, we show that the variance of the first passage time grows linearly with word distance along with word geodesic rays in every fixed boundary direction.

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