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CATEGORIES:Probability
SUMMARY:Geodesics in the Brownian map: Strong confluence a
nd geometric structure - Wei Qian (Orsay)
DTSTART;TZID=Europe/London:20210208T140000
DTEND;TZID=Europe/London:20210208T150000
UID:TALK157309AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/157309
DESCRIPTION:I will talk about our recent results on all geodes
ics in the Brownian map\, including those between
exceptional points. This is based on joint work wi
th Jason Miller (https://arxiv.org/abs/2008.02242)
.\n\nFirst\, we prove a strong and quantitative fo
rm of the confluence of geodesics phenomenon which
states that any pair of geodesics which are suffi
ciently close in the Hausdorff distance must coinc
ide with each other except near their endpoints.\n
\nThen\, we show that the intersection of any two
geodesics minus their endpoints is connected\, the
number of geodesics which emanate from a single p
oint and are disjoint except at their starting poi
nt is at most 5\, and the maximal number of geodes
ics which connect any pair of points is 9. For eac
h k=1\,…\,9\, we obtain the Hausdorff dimension of
the pairs of points connected by exactly k geodes
ics. For k=7\,8\,9\, such pairs have dimension zer
o and are countably infinite. Further\, we classif
y the (finite number of) possible configurations o
f geodesics between any pair of points\, up to hom
eomorphism\, and give a dimension upper bound for
the set of endpoints in each case. \n\nFinally\, w
e show that every geodesic can be approximated arb
itrarily well and in a strong sense by a geodesic
connecting typical points. In particular\, this g
ives an affirmative answer to a conjecture of Ange
l\, Kolesnik\, and Miermont that the geodesic fram
e\, the union of all of the geodesics in the Brown
ian map minus their endpoints\, has dimension one\
, the dimension of a single geodesic. \n
LOCATION:Zoom
CONTACT:Perla Sousi
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