  The task of calculating operator dimensions in the planar limit of N=4 super
Yang-Mills theory can be vastly simplified by mapping the dilatation generator
to the Hamiltonian of an integrable spin chain. The Bethe ansatz has been used
in this context to compute the spectra of spin chains associated with various
sectors of the theory which are known to decouple in the planar (large-N_c)
limit. These techniques are powerful at leading order in perturbation theory
but become increasingly complicated beyond one loop in the 't Hooft parameter
lambda=g_YM^2 N_c, where spin chains typically acquire long-range
(non-nearest-neighbor) interactions. In certain sectors of the theory,
moreover, higher-loop Bethe ansaetze do not even exist. We develop a virial
expansion of the spin chain Hamiltonian as an alternative to the Bethe ansatz
methodology, a method which simplifies the computation of dimensions of
multi-impurity operators at higher loops in lambda. We use these methods to
extract previously reported numerical gauge theory predictions near the BMN
limit for comparison with corresponding results on the string theory side of
the AdS/CFT correspondence. For completeness, we compare our virial results
with predictions that can be derived from current Bethe ansatz technology.
