A note on large deviations for the stable marriage of Poisson and Lebesgue with random appetites

TL;DR

This paper investigates large deviation probabilities for the distance between a typical point and its assigned center in a Poisson-Lebesgue stable marriage model with random appetites, extending previous results under moment restrictions.

Contribution

It generalizes existing large deviation results for the stable marriage problem to include random appetites with certain moment conditions.

Findings

Derived large deviation bounds for typical point distances

Extended previous models to incorporate random appetites

Provided conditions on appetite moments for the results

Abstract

Let $\Xi\subset\mathbb R^d$ be a set of centers chosen according to a Poisson point process in $\mathbb R^d$. Let $\psi$ be an allocation of $\mathbb R^d$ to $\Xi$ in the sense of the Gale-Shapley marriage problem, with the additional feature that every center $\xi\in\Xi$ has an appetite given by a nonnegative random variable $\alpha$. Generalizing some previous results, we study large deviations for the distance of a typical point $x\in\mathbb R^d$ to its center $\psi(x)\in\Xi$, subject to some restrictions on the moments of $\alpha$.