  We give asymptotic formulas for the multiplicities of weights and irreducible
summands in high-tensor powers $V_{\lambda}^{\otimes N}$ of an irreducible
representation $V_{\lambda}$ of a compact connected Lie group $G$. The weights
are allowed to depend on $N$, and we obtain several regimes of pointwise
asymptotics, ranging from a central limit region to a large deviations region.
We use a complex steepest descent method that applies to general asymptotic
counting problems for lattice paths with steps in a convex polytope.
