Rigorous results for two-dimensional random phonon systems

We present some rigorous sharp lower and upper bounds for the integrated phonon density of states and for the phonon specific heat of various types of two-dimensional disordered systems of masses and springs which are placed on triangular lattices. The bounds are given by quadrature formulas involving the first moments of the underlying probability distribution. We compare the results obtained for random systems up to the fourth order in moments with exact expressions corresponding to ordered systems with the same dimension and lattice structure.