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Almost sharp sharpness for Poisson Boolean percolation

Barbara Dembin, Vincent Tassion

2/9/22 Published in : arXiv:2209.00999

We consider Poisson Boolean percolation on \mathbb R^d with power-law distribution on the radius with a finite d-moment for d\ge 2. We prove that subcritical sharpness occurs for all but a countable number of power-law distributions. This extends the results of Duminil-Copin--Raoufi--Tassion where subcritical sharpness is proved under the assumption that the radii distribution has a 5d-3 finite moment. Our proofs techniques are different from their paper: we do not use randomized algorithm and rely on specific independence properties of Boolean percolation, inherited from the underlying Poisson process.
We also prove supercritical sharpness for any distribution with a finite d-moment and the continuity of the critical parameter for the truncated distribution when the truncation goes to infinity.

Entire article

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Locality of percolation for graphs with polynomial growth

Dynamics and 'arithmetics' of higher genus surface flows

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