Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Duality between star and plus connected components in percolation   Tile \(\mathbb{R}^2\) into disjoint unit squares \(\{S_k\}_{k \geq 0}\) with
the origin being the centre of \(S_0\) and say that \(S_i\) and \(S_j\) are
star adjacent if they share a corner and plus adjacent if they share an edge.
Every square is either vacant or occupied. Star and plus connected components
containing the origin have been previously studied using unicoherence and
interface graphs. In this paper, we use the structure of the outermost
boundaries derived in Ganesan (2015) to alternately obtain duality between star
and plus connected components in the following sense: There is a plus connected
cycle of vacant squares attached to surrounding the finite star connected
component containing the origin. There is a star connected cycle of vacant
squares attached to and surrounding the finite plus connected component
containing the origin.

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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
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The categories of this paper are: physics
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The categories of this paper are: cs
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The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-th
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The categories of this paper are: nlin