Zeros of a two-parameter random walk∗

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 N1+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 ((2pi)−1/2 + o(1)) logN almost surely