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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

OAI identifier:
oai:CiteSeerX.psu:10.1.1.313.7610

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