Abstract
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.
Citation
Piet Groeneboom. "Ulam’s Problem And Hammersley’s Process." Ann. Probab. 29 (2) 683 - 690, April 2001. https://doi.org/10.1214/aop/1008956689
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