Level crossings of a two-parameter random walk

We prove that the number Z(N) of level crossings of a two-parameter simple random walk in its first N×N steps is almost surely N3/2+o(1) as N→∞. The main ingredient is a strong approximation of Z(N) by the crossing local time of a Brownian sheet. Our result provides a useful algorithm for simulating the level sets of the Brownian sheet.