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

TL;DR

This paper investigates the percolation properties of a stable marriage model in continuous space where centers with random appetites are allocated regions, showing that small appetites prevent percolation under certain conditions.

Contribution

It generalizes previous models by incorporating random appetites and establishes conditions under which percolation does not occur.

Findings

Percolation is absent for small enough appetites.

The results depend on the moments of the random appetite distribution.

Extends prior work on stable allocations with deterministic appetites.

Abstract

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 $V$.