Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Critical random graphs: Diameter and mixing time   Let $\mathcal{C}_1$ denote the largest connected component of the critical
Erd\H{o}s--R\'{e}nyi random graph $G(n,{\frac{1}{n}})$. We show that,
typically, the diameter of $\mathcal{C}_1$ is of order $n^{1/3}$ and the mixing
time of the lazy simple random walk on $\mathcal{C}_1$ is of order $n$. The
latter answers a question of Benjamini, Kozma and Wormald. These results extend
to clusters of size $n^{2/3}$ of $p$-bond percolation on any $d$-regular
$n$-vertex graph where such clusters exist, provided that
$p(d-1)\le1+O(n^{-1/3})$.

26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph