Zeros of a two-parameter random walk

Details Zeros of a two-parameter random walk Zeros of a two-parameter random walk

2009-07-02

arxiv.org/abs/0907.0487v1

Mathematics

Probability

14 pages

Scientific paper

We prove that the number gamma(N) of the zeros of a two-parameter simple random walk in its first N-by-N time steps is almost surely equal to N to the power 1+o(1) as N goes to infinity. 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-by-N time steps is N to the power (3/2)+o(1) as N goes to infinity. We prove also that the number of zeros on the diagonal in the first N time steps is (c+o(1)) log N as N goes to infinity, where c is 2\pi.

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