Abstract
For two-dimensional last-passage time models of weakly increasing paths, interesting scaling limits have been proved for points close the axis (the hard edge). For strictly increasing paths of Bernoulli($p$) marked sites, the relevant boundary is the line $y=px$. We call this the soft edge to contrast it with the hard edge. We prove laws of large numbers for the maximal cardinality of a strictly increasing path in the rectangle $[p^{-1}n -xn^a]\times[n]$ as the parameters $a$ and $x$ vary. The results change qualitatively as $a$ passes through the value $1/2$.
Citation
Nicos Georgiou. "Soft edge results for longest increasing paths on the planar lattice." Electron. Commun. Probab. 15 1 - 13, 2010. https://doi.org/10.1214/ECP.v15-1519
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