  We consider corrections to the entropy of a black hole from an $O(N)$
invariant linear $\s$-model. We obtain the entropy from a $1/N$ expansion of
the partition function on a cone. The entropy arises from diagrams which are
analogous to those introduced by Susskind and Uglum to explain black hole
entropy in string theory. The interpretation of the \sm entropy depends on
scale. At short distances, it has a state counting interpretation, as the
entropy of entanglement of the $N$ fields $\pa$. In the infrared, the effective
theory has a single composite field $\s \sim \pa \pa$, and the state counting
interpretation of the entropy is lost.
