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 Π_t is the point process given by the center of the circles at time t, then, as t\to\infty, 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.

Fixed and extended proofs

References in corpus (1)

• Mobile Geometric Graphs, and Detection and Communication Problems in Mobile Wireless Networks

Cited by in corpus (2)

• Clustering and percolation of point processes
• Clustering, percolation and directionally convex ordering of point processes