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Zeros of a two-parameter random walk ∗

By Davar Khoshnevisan, Pál Révész and Technische Universität Wien


We prove that the number γN of the zeros of a two-parameter simple random walk in its first N ×N time steps is almost surely equal to N 1+o(1) as N → ∞. This is in contrast with our earlier joint effort with Z. Shi [4]; that work shows that the number of zero crossings in the first N × N time steps is N (3/2)+o(1) as N → ∞. We prove also that the number of zeros on the diagonal in the first N time steps is ((2π) −1/2 + o(1)) log N almost surely

Topics: Random walks, local time, random fields. AMS 2000 subject classification
Year: 2009
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