Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Upper tails and independence polynomials in random graphs   The upper tail problem in the Erd\H{o}s--R\'enyi random graph
$G\sim\mathcal{G}_{n,p}$ asks to estimate the probability that the number of
copies of a graph $H$ in $G$ exceeds its expectation by a factor $1+\delta$.
Chatterjee and Dembo showed that in the sparse regime of $p\to 0$ as
$n\to\infty$ with $p \geq n^{-\alpha}$ for an explicit $\alpha=\alpha_H>0$,
this problem reduces to a natural variational problem on weighted graphs, which
was thereafter asymptotically solved by two of the authors in the case where
$H$ is a clique. Here we extend the latter work to any fixed graph $H$ and
determine a function $c_H(\delta)$ such that, for $p$ as above and any fixed
$\delta>0$, the upper tail probability is $\exp[-(c_H(\delta)+o(1))n^2 p^\Delta
\log(1/p)]$, where $\Delta$ is the maximum degree of $H$. As it turns out, the
leading order constant in the large deviation rate function, $c_H(\delta)$, is
governed by the independence polynomial of $H$, defined as $P_H(x)=\sum i_H(k)
x^k $ where $i_H(k)$ is the number of independent sets of size $k$ in $H$. For
instance, if $H$ is a regular graph on $m$ vertices, then $c_H(\delta)$ is the
minimum between $\frac12 \delta^{2/m}$ and the unique positive solution of
$P_H(x) = 1+\delta$.

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The categories of this paper are: physics
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
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The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
27
The categories of this paper are: comp-gas