Edge mode amplification in disordered elastic networks.

Understanding how mechanical systems can be designed to efficiently transport elastic information is important in a variety of fields, including in materials science and biology. Recently, it has been discovered that certain crystalline lattices present "topologically-protected" edge modes that can amplify elastic signals. Several observations suggest that edge modes are important in disordered systems as well, an effect not well understood presently. Here we build a theory of edge modes in disordered isostatic materials and compute the distribution g(k) of Lyapunov exponents k characterizing how modes penetrate in the bulk, and find good agreement with numerical results. We show that disordered isostatic materials generically act as levers with amplification of an order L-L where L is the system size, whereas more connected materials amplify signals only close to free surfaces. Our approach, which is based on recent results in "free" random matrix theory, makes an analogy with electronic transport in a disordered conductor.