Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The complexity of spherical p-spin models - a second moment approach   Recently, Auffinger, Ben Arous, and \v{C}ern\'y initiated the study of
critical points of the Hamiltonian in the spherical pure $p$-spin spin glass
model, and established connections between those and several notions from the
physics literature. Denoting the number of critical values less than $Nu$ by
$\mbox{Crt}_{N}(u)$, they computed the asymptotics of
$\frac{1}{N}\log(\mathbb{E}\mbox{Crt}_{N}(u))$, as $N$, the dimension of the
sphere, goes to $\infty$. We compute the asymptotics of the corresponding
second moment and show that, for $p\geq3$ and sufficiently negative $u$, it
matches the first moment: \[ \mathbb{E}\left\{
\left(\mbox{Crt}_{N}\left(u\right)\right)^{2}\right\}
/\left(\vphantom{\left(\mbox{Crt}_{N}\left(u\right)\right)^{2}}\mathbb{E}\left\{
\mbox{Crt}_{N}\left(u\right)\right\} \right)^{2}\to1. \] As an immediate
consequence we obtain that $\mbox{Crt}_{N}(u)/\mathbb{E}\{ \mbox{Crt}_{N}(u)\}
\to 1$, in $L^2$ and thus in probability. For any $u$ for which
$\mathbb{E}\mbox{Crt}_{N}(u)$ does not tend to $0$ we prove that the moments
match on an exponential scale.

37
The categories of this paper are: supr-con
1
The categories of this paper are: cs
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th