### Local limit theorem on nilpotent Lie groups

Ergodic Theory and Dynamical Systems Seminar

1st December 2022, 2:00 pm – 3:00 pm

Fry Building, G07

We prove the local limit theorem for biased random walks on a simply connected nilpotent Lie group G. The result allows to approximate at scale 1 the n-step distribution of a walk by the time n of a smooth diffusion process for a new group structure on G. We also show this approximation is robust under deviation. The proof uses a Gaussian replacement scheme, combining Fourier analysis and a swapping argument inspired by the work of Diaconis-Hough. As a consequence, we obtain a probabilistic version of Ratner's theorem: Ad-unipotent random walks on finite-volume homogeneous spaces equidistribute toward algebraic measures.

This is joint work with Emmanuel Breuillard.

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