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SUMMARY:Dimension drop of the harmonic measure of some hyperbolic random w
alks
DTSTART;VALUE=DATE-TIME:20171204T153000Z
DTEND;VALUE=DATE-TIME:20171204T164500Z
DTSTAMP;VALUE=DATE-TIME:20210414T015312Z
UID:indico-event-2963@indico.math.cnrs.fr
DESCRIPTION:We consider the simple random walk on two types of tilings of
the hyperbolic plane. The first by 2π⁄q-angled regular polygons\, and t
he second by the Voronoi tiling associated to a random discrete set of the
hyperbolic plane\, the Poisson point process. In the second case\, we ass
ume that there are on average λ points per unit area.\n\nIn both cases th
e random walk (almost surely) escapes to infinity with positive speed\, an
d thus converges to a point on the circle. The distribution of this limit
point is called the harmonic measure of the walk.\n\nI will show that the
Hausdorff dimension of the harmonic measure is strictly smaller than 1\, f
or q sufficiently large in the Fuchsian case\, and for λ sufficiently sma
ll in the Poisson case. In particular\, the harmonic measure is singular w
ith respect to the Lebesgue measure on the circle in these two cases.\n\nT
he proof is based on a Furstenberg type formula for the speed together wit
h an upper bound for the Hausdorff dimension by the ratio between the entr
opy and the speed of the walk.\n\nThis is joint work with P. Lessa and E.
Paquette.\n\nhttps://indico.math.cnrs.fr/event/2963/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/2963/
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