Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non-perturbative corrections to mean-field behavior: spherical model on
  spider-web graph   We consider the spherical model on a spider-web graph. This graph is
effectively infinite-dimensional, similar to the Bethe lattice, but has loops.
We show that these lead to non-trivial corrections to the simple mean-field
behavior. We first determine all normal modes of the coupled springs problem on
this graph, using its large symmetry group. In the thermodynamic limit, the
spectrum is a set of $\delta$-functions, and all the modes are localized. The
fractional number of modes with frequency less than $\omega$ varies as $\exp
(-C/\omega)$ for $\omega$ tending to zero, where $C$ is a constant. For an
unbiased random walk on the vertices of this graph, this implies that the
probability of return to the origin at time $t$ varies as $\exp(- C' t^{1/3})$,
for large $t$, where $C'$ is a constant. For the spherical model, we show that
while the critical exponents take the values expected from the mean-field
theory, the free-energy per site at temperature $T$, near and above the
critical temperature $T_c$, also has an essential singularity of the type
$\exp[ -K {(T - T_c)}^{-1/2}]$.

28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
37
The categories of this paper are: supr-con
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg