Macroscopic wave propagation for 2D lattice with random masses

We consider a simple two-dimensional harmonic lattice with random, independent, and identically distributed masses. Using the methods of stochastic homogenization, we prove that solutions with initial data, which varies slowly relative to the lattice spacing, converge in an appropriate sense to solutions of an effective wave equation. The convergence is strong and almost sure. In addition, the role of the lattice's dimension in the rate of convergence is discussed. The technique combines energy estimates with powerful classical results about sub-Gaussian random variables.