Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gaussian fluctuations in mean field stable matchings

Daniel Ahlberg, Maria Deijfen, Tiffany Y. Y. Lo

TL;DR

This work analyzes stable matchings in a mean-field, complete bipartite setting with i.i.d. edge costs drawn from a continuous distribution $ ho$ possessing a near-zero density governed by a pseudo-dimension $d>0$. The authors show that the typical edge-cost in the stable matching scales as $n^{-1/d}$ with a universal limit, and that the total stable-matching cost exhibits dimension-dependent asymptotics: a law of large numbers for $d>1$ and a central limit theorem for $d>2$, with a precisely characterized variance constant $ ilde\,mbda(d)$ given by an integral expression. They prove these results first for Weibull$(d)$ costs and then extend to general $ ho$ via a quantile coupling, yielding robust universality of the results under the near-zero density condition. The methodology combines a martingale-difference approximation, a three-part decomposition of the total cost, and a careful transfer argument to handle general cost distributions, providing explicit limiting constants and insights into the regime where $d$ governs the scarcity of cheap edges. The findings illuminate how mean-field stable matchings behave under different tail regimes and connect to minimal matching results, with implications for understanding fluctuations in large random stable systems.

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.

Gaussian fluctuations in mean field stable matchings

TL;DR

This work analyzes stable matchings in a mean-field, complete bipartite setting with i.i.d. edge costs drawn from a continuous distribution possessing a near-zero density governed by a pseudo-dimension . The authors show that the typical edge-cost in the stable matching scales as with a universal limit, and that the total stable-matching cost exhibits dimension-dependent asymptotics: a law of large numbers for and a central limit theorem for , with a precisely characterized variance constant given by an integral expression. They prove these results first for Weibull costs and then extend to general via a quantile coupling, yielding robust universality of the results under the near-zero density condition. The methodology combines a martingale-difference approximation, a three-part decomposition of the total cost, and a careful transfer argument to handle general cost distributions, providing explicit limiting constants and insights into the regime where governs the scarcity of cheap edges. The findings illuminate how mean-field stable matchings behave under different tail regimes and connect to minimal matching results, with implications for understanding fluctuations in large random stable systems.

Abstract

Two sets of objects of size 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 satisfying as , where the number is known as the pseudo-dimension. For , the typical matching cost is shown to be of order , with an explicit distributional limit. For the total matching cost is shown to be of order , and to obey a law of large numbers. For the fluctuations of the total matching cost are shown to be of order , and to obey a central limit theorem.
Paper Structure (12 sections, 24 theorems, 180 equations, 2 figures)

This paper contains 12 sections, 24 theorems, 180 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Changing the scale parameter
  4. Idea of proofs
  5. Open problems
  6. Organisation of the paper
  7. The typical matching cost in the Weibull case
  8. A martingale difference approximation
  9. Variance of the total cost in the Weibull case
  10. LLN and CLT in the Weibull case
  11. The general case
  12. Technical proofs

Key Result

Theorem 1.1

Suppose that $\rho$ satisfies pd, for some $d>0$, and that $W$ satisfies Wlim. Then, $n^{1/d}c(v)\to W^{1/d}$ in distribution as $n\to\infty$. The convergence holds also in $L_p$, for $0< p<d$, if $\rho$ has finite $r$-th moment for some $r>2dp/(d-p)$. In addition, the $p$-th moment of $W^{1/d}$ is

Figures (2)

  • Figure 1: A numerical approximation of the integral $\gamma(d)$.
  • Figure 2: Simulations of $\widetilde{C}^\rho_{n,n}$ for $d=0.8$ (left) and $d=0.2$ (right), with $n = 10000$ and 5000 realisations for each $d$. The edge cost distribution is Weibull$(d)$ for either $d$.

Theorems & Definitions (47)

  • Theorem 1.1: The typical matching cost
  • Theorem 1.2: The total cost: Mean and LLN
  • Theorem 1.3: The total cost: Variance and CLT
  • Theorem 2.1: Typical cost in the Weibull case
  • Corollary 2.2: The mean of the total cost in the Weibull case
  • proof : Proof of Theorem \ref{['Ttyp']}
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Lemma 3.4
  • ...and 37 more