Perturbing the hexagonal circle packing

A percolation perspective

Itai Benjamini, A Stauffer

Research output: Contribution to journalArticle

2 Citations (Scopus)
70 Downloads (Pure)

Abstract

We consider the hexagonal circle packing with radius 1/2 and perturb it by letting the circles move as independent Brownian motions for time t. It is shown that, for large enough t, if Πt is the point process given by the center of the circles at time t, then, as t → ∞, the critical radius for circles centered at Πt to contain an infinite component converges to that of continuum percolation (which was shown - based on a Monte Carlo estimate - by Balister, Bollobás and Walters to be strictly bigger than 1/2). On the other hand, for small enough t, we show (using a Monte Carlo estimate for a fixed but high dimensional integral) that the union of the circles contains an infinite connected component. We discuss some extensions and open problems.
Original languageEnglish
Pages (from-to)1141-1157
Number of pages18
JournalAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
Volume49
Issue number4
DOIs
Publication statusPublished - Nov 2013

Fingerprint

Circle packing
Hexagon
Circle
Radius
Continuum Percolation
Point Process
Connected Components
Estimate
Brownian motion
Open Problems
Union
High-dimensional
Strictly
Converge
Integral
Point process

Cite this

Perturbing the hexagonal circle packing : A percolation perspective. / Benjamini, Itai; Stauffer, A.

In: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Vol. 49, No. 4, 11.2013, p. 1141-1157.

Research output: Contribution to journalArticle

@article{3fedd99d25aa4e9394eecc9f12548a4a,
title = "Perturbing the hexagonal circle packing: A percolation perspective",
abstract = "We consider the hexagonal circle packing with radius 1/2 and perturb it by letting the circles move as independent Brownian motions for time t. It is shown that, for large enough t, if Πt is the point process given by the center of the circles at time t, then, as t → ∞, the critical radius for circles centered at Πt to contain an infinite component converges to that of continuum percolation (which was shown - based on a Monte Carlo estimate - by Balister, Bollob{\'a}s and Walters to be strictly bigger than 1/2). On the other hand, for small enough t, we show (using a Monte Carlo estimate for a fixed but high dimensional integral) that the union of the circles contains an infinite connected component. We discuss some extensions and open problems.",
author = "Itai Benjamini and A Stauffer",
year = "2013",
month = "11",
doi = "10.1214/12-AIHP524",
language = "English",
volume = "49",
pages = "1141--1157",
journal = "Annales de l'Institut Henri Poincar{\'e}: Probabilit{\'e}s et Statistiques",
issn = "0246-0203",
publisher = "Institute of Mathematical Statistics",
number = "4",

}

TY - JOUR

T1 - Perturbing the hexagonal circle packing

T2 - A percolation perspective

AU - Benjamini, Itai

AU - Stauffer, A

PY - 2013/11

Y1 - 2013/11

N2 - We consider the hexagonal circle packing with radius 1/2 and perturb it by letting the circles move as independent Brownian motions for time t. It is shown that, for large enough t, if Πt is the point process given by the center of the circles at time t, then, as t → ∞, the critical radius for circles centered at Πt to contain an infinite component converges to that of continuum percolation (which was shown - based on a Monte Carlo estimate - by Balister, Bollobás and Walters to be strictly bigger than 1/2). On the other hand, for small enough t, we show (using a Monte Carlo estimate for a fixed but high dimensional integral) that the union of the circles contains an infinite connected component. We discuss some extensions and open problems.

AB - We consider the hexagonal circle packing with radius 1/2 and perturb it by letting the circles move as independent Brownian motions for time t. It is shown that, for large enough t, if Πt is the point process given by the center of the circles at time t, then, as t → ∞, the critical radius for circles centered at Πt to contain an infinite component converges to that of continuum percolation (which was shown - based on a Monte Carlo estimate - by Balister, Bollobás and Walters to be strictly bigger than 1/2). On the other hand, for small enough t, we show (using a Monte Carlo estimate for a fixed but high dimensional integral) that the union of the circles contains an infinite connected component. We discuss some extensions and open problems.

UR - http://www.scopus.com/inward/record.url?scp=84885088238&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1214/12-AIHP524

U2 - 10.1214/12-AIHP524

DO - 10.1214/12-AIHP524

M3 - Article

VL - 49

SP - 1141

EP - 1157

JO - Annales de l'Institut Henri Poincaré: Probabilités et Statistiques

JF - Annales de l'Institut Henri Poincaré: Probabilités et Statistiques

SN - 0246-0203

IS - 4

ER -