  Using exhaustion properties of invariant plurisubharmonic functions along
with basic combinatorial information on toric varieties convergence results for
sequences of distribution functions \phi_n=|s_N| / |s_N|_{L^2} for sections
s_N\in \Gamma (X,L^N) approaching a semiclassical ray are proved. Here X is a
normal compact toric variety and L is an ample line bundle equipped with an
arbitrary positive bundle metric which is invariant with respect to the compact
form of the torus.
