In 1977 Vershik and Kerov deduced the asymptotic normalized length of the longest increasing sequence among independent points uniformly distributed on the unit square. We solve the analogous problem for points on the planar square lattice that are present independently of each other.
"Increasing sequences of independent points on the planar lattice." Ann. Appl. Probab. 7 (4) 886 - 898, November 1997. https://doi.org/10.1214/aoap/1043862416