  We derive some new relationships between matrix models of Chern-Simons gauge
theory and of two-dimensional Yang-Mills theory. We show that q-integration of
the Stieltjes-Wigert matrix model is the discrete matrix model that describes
q-deformed Yang-Mills theory on the 2-sphere. We demonstrate that the
semiclassical limit of the Chern-Simons matrix model is equivalent to the
Gross-Witten model in the weak coupling phase. We study the strong coupling
limit of the unitary Chern-Simons matrix model and show that it too induces the
Gross-Witten model, but as a first order deformation of Dyson's circular
ensemble. We show that the Sutherland model is intimately related to
Chern-Simons gauge theory on the 3-sphere, and hence to q-deformed Yang-Mills
theory on the 2-sphere. In particular, the ground state wavefunction of the
Sutherland model in its classical equilibrium configuration describes the
Chern-Simons free energy. The correspondence is extended to Wilson line
observables and to arbitrary simply-laced gauge groups.
