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

Daniel Andr\'es D\'iaz-Pach\'on
Arxiv ID: 0909.5325Last updated: 11/16/2021
Let $\Xi$ be a set of centers chosen according to a Poisson point process in $\mathbb R^d$. Consider the allocation of $\mathbb R^d$ to $\Xi$ which is stable in the sense of the Gale-Shapley marriage problem, with the additional feature that every center $\xi\in\Xi$ has a random appetite $\alpha V$, where $\alpha$ 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 $\alpha$ is small enough, depending on certain characteristics of the moment of $V$.

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