Longest increasing subsequences for distributions with atoms, and an inhomogeneous Hammersley process

A famous result by Hammersley and Versik-Kerov states that the length $L_n$ of the longest increasing subsequence among $n$ iid continuous random variables grows like $2\sqrt{n}$. We investigate here the asymptotic behavior of $L_n$ for distributions with atoms. For purely discrete random variables, we characterize the asymptotic order of $L_n$ through a variational problem and provide explicit estimates for classical distributions. The proofs rely on a coupling with an inhomogeneous version of the discrete-time continuous-space Hammersley process. This reveals that, in contrast to the continuous case, the discrete setting exhibits a wide range of growth rates between $\mathcal{O}(1)$ and $o(\sqrt{n})$, depending on the tail behavior of the distribution. We can then easily deduce the asymptotics of $L_n$ for a completely arbitrary distribution.