Ulam’s Problem And Hammersley’s Process

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.