Patience Sorting, Longest Increasing Subsequences and a Continuous Space Analog of the Simple Asymmetric Exclusion Process

There is a broad circle of ideas relating to the length Ln of the longest increasing subsequence of a random n-permutation. After reviewing known results and methods, we develop two new themes. The simplest algorithm for computing the length of the longest increasing subsequence can be viewed as a card gard, patience sorting, and one theme is to give the first asymptotic probabilistic analysis of this game. The second theme is that a continuous limit process, which we call Hammersley’s process, exists as a continuous-space interacting particle process. This turns out to be analogous to the (discrete-space) simple asymmetric exclusion process, and also to be an elaboration of Hammersley’s Poisson process representation. The celebrated result ELn ∼ 2n is intimately tied to the hydrodynamical limit theorem for Hammersley’s process. ∗Research supported by N.S.F. Grant MCS 92-24857