Jerusalem Mathematics Colloquium

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Thursday, 9th November 2000, 4:00 pm
Mathematics Bldg., lecture hall 2

Professor Ofer Zeitouni
(Department of Electrical Engineering, Technion)

"Frequent points and covering times:
On conjectures of Erdos-Taylor and of Kesten-Revesz"


Consider the simple random walk in Z^2. Let T_n denote the time spent at the most frequently visited site by time n, and let C_n denote the time needed to cover completely the disc of radius n. Erdos and Taylor (1960) proved that

1/(4 pi) <= liminf_{n -> oo} T_n/(log n)^2 <= limsup_{n -> oo} T_n/(log n)^2 <= 1/pi, a.s.

and conjectured that the upper bound is sharp. Kesten and Revesz (separately, around 1990) proved that

e^{-4/t} <= liminf_{n -> oo} P[ log C_n <= t (log n)^2 ]
<= limsup_{n -> oo} P[ log C_n) <= t(log n)^2 ] <= e^{-1/t}

and conjectured the lower bound is sharp. The latter probability estimate is strongly related to limits of the cover time of the discrete two dimensional torus by the simple random walk. We will resolve affirmatively both conjectures, studying first Brownian motion in R^2 and on the two dimensional torus. Our approach is strongly motivated by similar questions on trees. As a byproduct, one obtains an understanding of how is the most visited site hit and what is the structure of visits around the last point to be covered in the disc.

(Joint work with Amir Dembo, Yuval Peres and Jay Rosen.)

Coffee, Cookies at the faculty lounge at 3:30.

You are invited to join the speaker for further discussion after the talk at Beit Belgia.

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