  We determine the asymptotics of the joint eigenfunctions of the torus action
on a toric Kahler variety. Such varieties are models of completely integrable
systems in complex geometry. We first determine the pointwise asymptotics of
the eigenfunctions, which show that they behave like Gaussians centered at the
corresponding classical torus. We then show that there is a universal Gaussian
scaling limit of the distribution function near its center. We also determine
the limit distribution for the tails of the eigenfunctions on large length
scales. These are not universal but depend on the global geometry of the toric
variety and in particular on the details of the exponential decay of the
eigenfunctions away from the classically allowed set.
