Regularity of the time constant for last passage percolation on complete directed acyclic graphs

We study the time constant $C(\nu )$ of last passage percolation on the complete directed acyclic graph on the set of non-negative integers, where edges have i.i.d. weights with distribution $\nu$ with support included in $\{-\mathrm{\infty }\}\cup \mathbb{R}$. We show that $\nu \mapsto C(\nu )$ is strictly increasing in $\nu$. We also prove that $C(\nu )$ is continuous in $\nu$ for a large set of measures $\nu$. Furthermore, when $\nu$ is purely atomic, we show that $C(\nu )$ is analytic with respect to the weights of the atoms. In the special case of two positive atoms, it is an explicit rational function of these weights.