Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Maker-Breaker Percolation Games II: Escaping to Infinity   Let $\Lambda$ be an infinite connected graph, and let $v_0$ be a vertex of
$\Lambda$. We consider the following positional game. Two players, Maker and
Breaker, play in alternating turns. Initially all edges of $\Lambda$ are marked
as unsafe. On each of her turns, Maker marks $p$ unsafe edges as safe, while on
each of his turns Breaker takes $q$ unsafe edges and deletes them from the
graph. Breaker wins if at any time in the game the component containing $v_0$
becomes finite. Otherwise if Maker is able to ensure that $v_0$ remains in an
infinite component indefinitely, then we say she has a winning strategy. This
game can be thought of as a variant of the celebrated Shannon switching game.
Given $(p,q)$ and $(\Lambda, v_0)$, we would like to know: which of the two
players has a winning strategy?
  Our main result in this paper establishes that when $\Lambda = \mathbb{Z}^2$
and $v_0$ is any vertex, Maker has a winning strategy whenever $p\geq 2q$,
while Breaker has a winning strategy whenever $2p\leq q$. In addition, we
completely determine which of the two players has a winning strategy for every
pair $(p,q)$ when $\Lambda$ is an infinite $d$-regular tree. Finally, we give
some results for general graphs and lattices and pose some open problems.

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The categories of this paper are: acc-phys
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The categories of this paper are: cs
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph