Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the size of a finite vacant cluster of random interlacements with
  small intensity   In this paper we establish some properties of percolation for the vacant set
of random interlacements, for d at least 5 and small intensity u. The model of
random interlacements was first introduced by A.S. Sznitman in arXiv:0704.2560.
It is known that, for small u, almost surely there is a unique infinite
connected component in the vacant set left by the random interlacements at
level u, see arXiv:0808.3344 and arXiv:0805.4106. We estimate here the
distribution of the diameter and the volume of the vacant component at level u
containing the origin, given that it is finite. This comes as a by-product of
our main theorem, which proves a stretched exponential bound on the probability
that the interlacement set separates two macroscopic connected sets in a large
cube. As another application, we show that with high probability, the unique
infinite connected component of the vacant set is `ubiquitous' in large
neighborhoods of the origin.

31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio