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Random matching in 2D with exponent 2 for gaussian densities

Emanuele Caglioti, Francesca Pieroni

TL;DR

The Random Euclidean Matching problem with exponent 2 for the Gaussian distribution defined on the plane is solved for the Gaussian distribution defined on the plane using a geometric decomposition allowing an explicit computation of the constant.

Abstract

We solve the Random Euclidean Matching problem with exponent 2 for the Gaussian distribution defined on the plane. Previous works by Ledoux and Talagrand determined the leading behavior of the average cost up to a multiplicative constant. We explicitly determine the constant, showing that the average cost is proportional to (log N)^2, where N is the number of points. Our approach relies on a geometric decomposition allowing an explicit computation of the constant. Our results illustrate the potential for exact solutions of random matching problems for many distributions defined on unbounded domains on the plane.

Random matching in 2D with exponent 2 for gaussian densities

TL;DR

The Random Euclidean Matching problem with exponent 2 for the Gaussian distribution defined on the plane is solved for the Gaussian distribution defined on the plane using a geometric decomposition allowing an explicit computation of the constant.

Abstract

We solve the Random Euclidean Matching problem with exponent 2 for the Gaussian distribution defined on the plane. Previous works by Ledoux and Talagrand determined the leading behavior of the average cost up to a multiplicative constant. We explicitly determine the constant, showing that the average cost is proportional to (log N)^2, where N is the number of points. Our approach relies on a geometric decomposition allowing an explicit computation of the constant. Our results illustrate the potential for exact solutions of random matching problems for many distributions defined on unbounded domains on the plane.
Paper Structure (9 sections, 20 theorems, 155 equations, 6 figures)

This paper contains 9 sections, 20 theorems, 155 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Useful results and notations
  4. A piecewise multiscaling density
  5. The Gaussian density
  6. Preliminary estimates
  7. Convergence Theorems
  8. The Maxwellian density
  9. Appendix

Key Result

Lemma 2.1

Let $\rho$ be any probability density on $\mathbb{R}^2$ and let $X_1,\dots,X_N$ and $Y_1,\dots,Y_N$ independent random variables in $\mathbb{R}^2$ with common distribution $\rho$, then

Figures (6)

  • Figure 1.1: Multiscaling density
  • Figure 3.1: Multiscaling density
  • Figure 4.1: The set of squares $Q^j_k$ where the cut-off is applied.
  • Figure 4.2: The set of rectangles where the cut-off is applied, except for zero measure sets.
  • Figure 4.3: A graphical representation of the proof. $N^j_k$ is the number of particles in $Q^j_k$ (the first square in blue), while $P^j_k$ is the number of particles in the red rectangle. Once fixed $N_k$, that is the number of particles in $R_k$, the fluctuations of the particles in the red rectangle are exactly the fluctuations of the particles in the blue one.
  • ...and 1 more figures

Theorems & Definitions (31)

  • Lemma 2.1: AST, Proposition 4.8
  • Theorem 2.1: BB, Benamou-Brenier formula
  • Theorem 2.2: AG, Corollary 4.4
  • Theorem 2.3: T1996, Talagrand formula
  • Theorem 2.4: AGT, Theorems 1.1 & 1.2, Propositions 3.1 & 3.2
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Proposition 3.1
  • proof
  • ...and 21 more