An afternoon on random walks and groups

Abstract: 15.30 - 16.15 Giulio Tiozzo (Toronto)

16.30 - 17.15 Sébastien Gouëzel (Rennes)

17.45 - 18.30 Andrei Alpeev (St-Petersburg)

Giulio Tiozzo, "The fundamental inequality for cocompact Fuchsian groups".

A recurring question in the theory of random walks on hyperbolic spaces asks whether the hitting (harmonic) measures can coincide with measures of geometric origin, such as the Lebesgue measure. This is also related to the inequality between entropy and drift. For finitely-supported random walks on cocompact Fuchsian groups with symmetric fundamental domain, we prove that the hitting measure is singular with respect to Lebesgue measure; moreover, its Hausdorff dimension is strictly less than 1. Along the way, we prove a purely geometric inequality for geodesic lengths, strongly reminiscent of the Anderson-Canary-Culler-Shalen inequality for free Kleinian groups. Joint with P. Kosenko.

Sébastien Gouëzel, "Exponential estimates for random walks without moment conditions on hyperbolic spaces"

Consider a random walk on a nonelementary hyperbolic space (proper or not, but one may just think of a free group for simplicity). It is known that the walk is converging almost surely towards a point at a boundary, and that the rate of escape is positive. We will discuss quantitative versions of these statements: when can one show that these facts hold with an exponentially small probability for exceptions? While there are several such results in the literature, the originality of our approach is that it does not require any moment condition on the random walk. We will discuss the main technical new idea in the case of the free group.

Andrei Alpeev, "Examples of different boundary behaviour of left and right random walks on groups".

group theory

Audience: researchers in the topic

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ENS group theory seminar

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