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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
MersenneTwister
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