Macroscopic Wave Propagation for 2D Lattice with Random Masses

TL;DR

This paper demonstrates that in a 2D harmonic lattice with random masses, solutions with long wave initial data converge strongly and almost surely to an effective wave equation, highlighting the impact of lattice dimension on convergence rate.

Contribution

It applies stochastic homogenization techniques to prove strong, almost sure convergence of solutions in a 2D lattice with random masses, revealing the influence of dimension on convergence speed.

Findings

Solutions converge to an effective wave equation

Convergence is strong and almost sure

Lattice dimension affects convergence rate

Abstract

We consider a simple two-dimemsional harmonic lattice with random, independent and identically distributed masses. Using the methods of stochastic homogenization, we show that solutions with long wave initial data 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.