Gaussian and non-Gaussian fluctuations in mean field stable matchings

Abstract

Two sets of objects of size $n$ are to be matched to each other based on i.i.d.\ costs associated to every pair of objects. Objects prefer to be matched as cheaply as possible, and a matching is said to be stable if there is no pair of objects that would prefer to match to each other rather than to their current partners. Properties of such matchings are analysed for cost distributions with a density $ρ$ satisfying $ρ(x)/(dx^{d-1})\to 1$ as $x\to 0^+$, where the number $d$ is known as the pseudo-dimension. For $d>0$, the typical matching cost is shown to be of order $n^{-1/d}$, with an explicit distributional limit. For $d>1$ the total matching cost is shown to be of order $n^{1-1/d}$, and to obey a law of large numbers. For $d> 2$, the fluctuations of the total matching cost are shown to be of order $n^{1/2-1/d}$, and to obey a central limit theorem. In contrast, for $1<d<2$ fluctuations are non-Gaussian and of constant order, whereas at the critical value $d=2$ fluctuations are shown to be order $\sqrt{\log n}$ and occasionally of Gaussian nature.