Soft edge results for longest increasing paths on the planar lattice

Details Soft edge results for longest increasing paths on the planar lattice Soft edge results for longest increasing paths on the planar lattice

2010-03-17

arxiv.org/abs/1003.3286v1

Electronic Communications in Probability, Volume 15(2010), pages 1-13

Mathematics

Probability

14 pages, 2 figures

Scientific paper

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 $[\fl{p^{-1}n -xn^a}]\times[n]$ as the parameters $a$ and $x$ vary. The results change qualitatively as $a$ passes through the value 1/2.

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