Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharpness of the phase transition for continuum percolation in R^2   We study the phase transition of random radii Poisson Boolean percolation:
Around each point of a planar Poisson point process, we draw a disc of random
radius, independently for each point. The behavior of this process is well
understood when the radii are uniformly bounded from above. In this article, we
investigate this process for unbounded (and possibly heavy tailed) radii
distributions. Under mild assumptions on the radius distribution, we show that
both the vacant and occupied sets undergo a phase transition at the same
critical parameter $\lambda_c$. Moreover,
  - For $\lambda < \lambda_c$, the vacant set has a unique unbounded connected
component and we give precise bounds on the one-arm probability for the
occupied set, depending on the radius distribution.
  - At criticality, we establish the box-crossing property, implying that no
unbounded component can be found, neither in the occupied nor the vacant sets.
We provide a polynomial decay for the probability of the one-arm events, under
sharp conditions on the distribution of the radius.
  - For $\lambda > \lambda_c$, the occupied set has a unique unbounded
component and we prove that the one-arm probability for the vacant decays
exponentially fast.
  The techniques we develop in this article can be applied to other models such
as the Poisson Voronoi and confetti percolation.

24
The categories of this paper are: dg-ga
22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
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The categories of this paper are: atom-ph
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph