Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The phase transition in site percolation on pseudo-random graphs   We establish the existence of the phase transition in site percolation on
pseudo-random $d$-regular graphs. Let $G=(V,E)$ be an $(n,d,\lambda)$-graph,
that is, a $d$-regular graph on $n$ vertices in which all eigenvalues of the
adjacency matrix, but the first one, are at most $\lambda$ in their absolute
values. Form a random subset $R$ of $V$ by putting every vertex $v\in V$ into
$R$ independently with probability $p$. Then for any small enough constant
$\epsilon>0$, if $p=\frac{1-\epsilon}{d}$, then with high probability all
connected components of the subgraph of $G$ induced by $R$ are of size at most
logarithmic in $n$, while for $p=\frac{1+\epsilon}{d}$, if the eigenvalue ratio
$\lambda/d$ is small enough as a function of $\epsilon$, then typically $R$
spans a connected component of size at least $\frac{\epsilon n}{d}$ and a path
of length proportional to $\frac{\epsilon^2n}{d}$.

14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci