  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.
