Subcritical phase of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> -dimensional Poisson–Boolean percolation and its vacant set
Hugo Duminil‐Copin, Aran Raoufi, Vincent Tassion
Abstract
We prove that the Poisson–Boolean percolation on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>ℝ</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> undergoes a sharp phase transition in any dimension under the assumption that the radius distribution has a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>5</mml:mn> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> finite moment (in particular we do not assume that the distribution is bounded). To the best of our knowledge, this is the first proof of sharpness for a model in dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> that does not exhibit exponential decay of connectivity probabilities in the subcritical regime. More precisely, we prove that in the whole subcritical regime, the expected size of the cluster of the origin is finite, and furthermore we obtain bounds for the origin to be connected to distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> : when the radius distribution has a finite exponential moment, the probability decays exponentially fast in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> , and when the radius distribution has heavy tails, the probability is equivalent to the probability that the origin is covered by a ball going to distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> (this result is new even in two dimensions). In the supercritical regime, it is proved that the probability of the origin being connected to infinity satisfies a mean-field lower bound. The same proof carries on to conclude that the vacant set of Poisson–Boolean percolation on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>ℝ</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> undergoes a sharp phase transition.