Perturbing the hexagonal circle packing: A percolation perspective

Itai Benjamini, A Stauffer

Research output: Contribution to journalArticle

3 Citations (Scopus)
127 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

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