Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Eigenvalues outside the bulk of inhomogeneous Erd\H{o}s-R\"enyi random
  graphs   The article considers an inhomogeneous Erd\H{o}s-R\"enyi random graph on
$\{1,\ldots, N\}$, where an edge is placed between vertices $i$ and $j$ with
probability $\varepsilon_N f(i/N,j/N)$, for $i\le j$, the choice being made
independent for each pair. The function $f$ is assumed to be non-negative
definite, symmetric, bounded and of finite rank $k$. We study the edge of the
spectrum of the adjacency matrix of such an inhomogeneous Erd\H{o}s-R\'enyi
random graph under the assumption that $N\varepsilon_N\to \infty$ sufficiently
fast. Although the bulk of the spectrum of the adjacency matrix, scaled by
$\sqrt{N\varepsilon_N}$, is compactly supported, the $k$-th largest eigenvalue
goes to infinity. It turns out that the largest eigenvalue after appropriate
scaling and centering converge to a Gaussian law, if the largest eigenvalue of
$f$ has multiplicity $1$. If $f$ has $k$ distinct non-zero eigenvalues, then
the joint distribution of the $k$ largest eigenvalues converge jointly to a
multivariate Gaussian law. The first order behaviour of the eigenvectors is
derived as a by-product of the above results. The results complement the
homogeneous case derived by Erd\H{o}s et al.(2013).

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The categories of this paper are: q-alg
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-fin
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The categories of this paper are: eess
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The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
3
The categories of this paper are: physics
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The categories of this paper are: nucl-th
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The categories of this paper are: econ
6
The categories of this paper are: astro-ph
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The categories of this paper are: math-ph