Regularity of the time constant for last passage percolation on complete directed acyclic graphs
Abstract
We study the time constant C(ν) 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 ν with support included in {−∞} ∪ R. We show that ν → C(ν) is strictly increasing in ν. We also prove that C(ν) is continuous in ν for a large set of measures ν. Furthermore, when ν is purely atomic, we show that C(ν) 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.
