Source: Ann. Probab. Volume 29, Number 2
(2001), 683-690.
Let $L_n$ be the length of the longest increasing subsequence of a
random permutation of the numbers $1,\ldots, n$, for the uniform distribution
on the set of permutations. Hammersley's interacting particle process, implicit
in Hammersley (1972), has been used in Aldous and Diaconis (1995) to provide a
“soft” hydrodynamical argument for proving that $\lim_{n \to
\infty} EL_n / \sqrt{n} = 2$. We show in this note that the latter result is in
fact an immediate consequence of properties of a random 2dimensional
signed measure, associated with Hammersley’s process.
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