@article{me71,
	author="D.J. Aldous and P. Diaconis",
	title="Hammersley's Interacting Particle Process and Longest
Increasing Subsequences",
	journal="Probab. Th. Rel. Fields",
	volume=103,
	pages="199-213",
	year=1995}

(Abstract)
In a famous paper Hammersley (1970)
investigated the length $L_n$ of the longest increasing subsequence
of a random $n$-permutation.
Implicit in that paper is a certain
one-dimensional continuous-space interacting particle process.
By studying a hydrodynamical limit for Hammersley's process we
show by fairly ``soft" arguments that 
$\lim n^{-1/2} EL_n = 2$.
This is a known result, but previous proofs
(Vershik - Kerov  (1977); Logan - Shepp (1977))
relied on hard analysis of combinatorial asymptotics.