  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
$\exp[ -K {(T - T_c)}^{-1/2}]$.
