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Perturbing the hexagonal circle packing: a percolation perspective. (English. French summary) Zbl 1383.60094
Summary: 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 \(t\to\infty\), 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 P. Balister et al. [Random Struct. Algorithms 26, No. 4, 392–403 (2005; Zbl 1072.60083)] 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.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C43 Time-dependent percolation in statistical mechanics
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
52C26 Circle packings and discrete conformal geometry
Software:
MersenneTwister
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