Perturbing the hexagonal circle packing: a percolation perspective

Itai Benjamini; Alexandre Stauffer

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 4, page 1141-1157
  • ISSN: 0246-0203

Abstract

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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 𝛱 t is the point process given by the center of the circles at time t , then, as t → ∞ , the critical radius for circles centered at 𝛱 t 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.

How to cite

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Benjamini, Itai, and Stauffer, Alexandre. "Perturbing the hexagonal circle packing: a percolation perspective." Annales de l'I.H.P. Probabilités et statistiques 49.4 (2013): 1141-1157. <http://eudml.org/doc/272086>.

@article{Benjamini2013,
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 $\varPi _\{t\}$ is the point process given by the center of the circles at time $t$, then, as $t\rightarrow \infty $, the critical radius for circles centered at $\varPi _\{t\}$ 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.},
author = {Benjamini, Itai, Stauffer, Alexandre},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {hexagonal circle packing; brownian motion; continuum percolation; Brownian motion},
language = {eng},
number = {4},
pages = {1141-1157},
publisher = {Gauthier-Villars},
title = {Perturbing the hexagonal circle packing: a percolation perspective},
url = {http://eudml.org/doc/272086},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Benjamini, Itai
AU - Stauffer, Alexandre
TI - Perturbing the hexagonal circle packing: a percolation perspective
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 4
SP - 1141
EP - 1157
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 $\varPi _{t}$ is the point process given by the center of the circles at time $t$, then, as $t\rightarrow \infty $, the critical radius for circles centered at $\varPi _{t}$ 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.
LA - eng
KW - hexagonal circle packing; brownian motion; continuum percolation; Brownian motion
UR - http://eudml.org/doc/272086
ER -

References

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