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For two-dimensional last-passage time models of weakly increasing paths,interesting scaling limits have been proved for points close the axis (the hardedge). For strictly increasing paths of Bernoulli($p$) marked sites, therelevant boundary is the line $y=px$. We call this the soft edge to contrast itwith the hard edge. We prove laws of large numbers for the maximal cardinalityof a strictly increasing path in the rectangle $[\fl{p^{-1}n -xn^a}]\times[n]$as the parameters $a$ and $x$ vary. The results change qualitatively as $a$passes through the value 1/2.

Soft edge results for longest increasing paths on the planar lattice