Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Upper tail of the spectral radius of sparse Erd\H{o}s-R\'{e}nyi graphs   We consider an Erd\H{o}s-R\'{e}nyi graph $\mathbb{G}(n,p)$ on $n$ vertices
with edge probability $p$ such that \[ \sqrt{\frac{\log n}{\log \log n}} \ll np
\le n^{1/2-o(1)}, \label{eq:abs} \tag{$\dagger$} \] and derive the upper tail
large deviations of $\lambda(\mathbb{G}(n,p))$, the largest eigenvalue of its
adjacency matrix. Within this regime we show that, for $p \gg n^{-2/3}$ the
$\log$-probability of the upper tail event of $\lambda(\mathbb{G}(n,p))$ equals
to that of planting a clique of an appropriate size (upon ignoring smaller
order terms), while for $p \ll n^{-2/3}$ the same is given by that of the
existence of a high degree vertex. This, in particular, shows an emergence of
{\em non-planted localized structure} in the latter regime. We also confirm
that in the entire regime \eqref{eq:abs} the large deviation probability is
asymptotically approximated by the solution of the mean-field variational
problem, and further identify the typical structure of $\mathbb{G}(n,p)$
conditioned on the upper tail event of $\lambda(\mathbb{G}(n,p))$ in a certain
sub-regime of $p$.
  For $p$ such that $\log(np) \gtrsim \log n$ the large deviations of
$\lambda(\mathbb{G}(n,p))$ is deduced from those of ${\rm Hom}(C_{2t},
\mathbb{G}(n,p))$, the homomorphism counts of cycle of length $2t$, for $t \ge
3$ and $p$ such that $n^{1/2-o(1)} \ge np \gg n^{1/t}$. In this latter regime
the typical structure of $\mathbb{G}(n,p)$ conditioned on the upper tail of
${\rm Hom}(C_{2t}, \mathbb{G}(n,p))$ is identified and asymptotic tightness of
the mean-field approximation is also established.

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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: cs
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The categories of this paper are: hep-th
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The categories of this paper are: solv-int
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The categories of this paper are: quant-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: physics
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The categories of this paper are: alg-geom