# Perturbing the hexagonal circle packing

## A percolation perspective

Itai Benjamini, A Stauffer

Research output: Contribution to journalArticle

2 Citations (Scopus)

### 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 language English 1141-1157 18 Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 49 4 https://doi.org/10.1214/12-AIHP524 Published - Nov 2013

### Fingerprint

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

### Cite this

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

Research output: Contribution to journalArticle

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