Open Access
2010 Soft edge results for longest increasing paths on the planar lattice
Nicos Georgiou
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Electron. Commun. Probab. 15: 1-13 (2010). DOI: 10.1214/ECP.v15-1519


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 $1/2$.


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Nicos Georgiou. "Soft edge results for longest increasing paths on the planar lattice." Electron. Commun. Probab. 15 1 - 13, 2010.


Accepted: 7 January 2010; Published: 2010
First available in Project Euclid: 6 June 2016

zbMATH: 1202.60154
MathSciNet: MR2581043
Digital Object Identifier: 10.1214/ECP.v15-1519

Primary: 60K35

Keywords: Bernoulli matching model , Discrete TASEP , increasing paths , last passage model , soft edge , Weak law of large numbers

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