Soft edge results for longest increasing paths on the planar lattice

TL;DR

This paper investigates the asymptotic behavior of the longest strictly increasing paths in a planar lattice, introducing the concept of a 'soft edge' and analyzing how the path length scales near this boundary.

Contribution

It establishes laws of large numbers for the maximal path length in specific rectangles, revealing qualitative changes as a parameter crosses 1/2.

Findings

Law of large numbers for path length as parameters vary

Identification of a phase transition at parameter value 1/2

Analysis of soft edge boundary effects in lattice paths

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