Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotics in percolation on high-girth expanders   We consider supercritical bond percolation on a family of high-girth
$d$-regular expanders. Alon, Benjamini and Stacey (2004) established that its
critical probability for the appearance of a linear-sized ("giant'') component
is $p_c=1/(d-1)$. Our main result recovers the sharp asymptotics of the size
and degree distribution of the vertices in the giant and its 2-core at any
$p>p_c$. It was further shown in [ABS04] that the second largest component, at
any $0<p<1$, has size at most $n^{\omega}$ for some $\omega<1$. We show that,
unlike the situation in the classical Erd\H{o}s-R\'enyi random graph, the
second largest component in bond percolation on a regular expander, even with
an arbitrarily large girth, can have size $n^{\omega'}$ for $\omega'$
arbitrarily close to $1$. Moreover, as a by-product of that construction, we
answer negatively a question of Benjamini (2013) on the relation between the
diameter of a component in percolation on expanders and the existence of a
giant component. Finally, we establish other typical features of the giant
component, e.g., the existence of a linear path.

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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: acc-phys
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The categories of this paper are: solv-int
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: cs
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: math-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ph