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UID:7978@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20151106T110000
DTEND;TZID=Europe/Paris:20151106T120000
DTSTAMP:20241120T205618Z
URL:https://www.i2m.univ-amu.fr/evenements/the-hunter-cauchy-rabbit-and-op
 timal-kakeya-set-bruno-schapira/
SUMMARY:Bruno Schapira (I2M\, Aix-Marseille Université): The hunter\, Cauc
 hy Rabbit\, and optimal Kakeya set - Bruno Schapira
DESCRIPTION:Bruno Schapira: "A planar set that contains a unit segment in e
 very direction is called a Kakeya set. These sets have been studied intens
 ively in geometric measure theory and harmonic analysis since the work of 
 Besicovich (1928)\; we find a new connection to game theory and probabilit
 y. A hunter and a rabbit move on the integer points in [0\,n) without seei
 ng each other. At each step\, the hunter moves to a neighboring vertex or 
 stays in place\, while the rabbit is free to jump to any node. Thus they a
 re engaged in a zero sum game\, where the payoff is the capture time. The 
 known optimal randomized strategies for hunter and rabbit achieve expected
  capture time of order n log n. We show that every rabbit strategy yields 
 a Kakeya set\; the optimal rabbit strategy is based on a discretized Cauch
 y random walk\, and it yields a Kakeya set K consisting of 4n triangles\, 
 that has minimal area among such sets (the area of K is of order 1/log(n))
 . Passing to the scaling limit yields a simple construction of a random Ka
 keya set with zero area from two Brownian motions."\nA work by Y. Babichen
 ko\, Y. Peres\, R. Peretz\, P. Sousi and P. Winkler.\n\nReference: https:/
 /arxiv.org/abs/1207.6389\n\n&nbsp\;
ATTACH;FMTTYPE=image/jpeg:https://www.i2m.univ-amu.fr/wp-content/uploads/2
 015/11/Bruno_Schapira.jpg
CATEGORIES:Séminaire,Probabilités
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