Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence of an unbounded vacant set for subcritical continuum
  percolation   We consider the Poisson Boolean percolation model in $\mathbb{R}^2$, where
the radii of each ball is independently chosen according to some probability
measure with finite second moment. For this model, we show that the two
thresholds, for the existence of an unbounded occupied and an unbounded vacant
component, coincide. This complements a recent study of the sharpness of the
phase transition in Poisson Boolean percolation by the same authors. As a
corollary it follows that for Poisson Boolean percolation in $\mathbb{R}^d$,
for any $d\ge2$, finite moment of order $d$ is both necessary and sufficient
for the existence of a nontrivial phase transition for the vacant set.

35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess