Percolation for the stable marriage of Poisson and Lebesgue with random appetites

Abstract

Let $Ξ$ be a set of centers chosen according to a Poisson point process in $\mathbb R^d$. Consider the allocation of $\mathbb R^d$ to $Ξ$ which is stable in the sense of the Gale-Shapley marriage problem, with the additional feature that every center $ξ\inΞ$ has a random appetite $αV$, where $α$ is a nonnegative scale constant and $V$ is a nonnegative random variable. Generalizing previous results by Freire, Popov and Vachkovskaia (\cite{FPV}), we show the absence of percolation when $α$ is small enough, depending on certain characteristics of the moment of $V$.