Scaling of energy spreading in a disordered Ding-Dong lattice
We study numerically propagation of energy in a one dimensional Ding-Ding lattice, composed of linear oscillators with ellastic collisions. Wave propagation is suppressed by breaking translational symmetry, we consider three way to do this: a position disorder, a mass disorder, and a dimer lattice with alternating distances between the units. In all cases the spreading of an initially localized wavepacket is irregular, due to appearance of chaos, and subdiffusive. Guided by a nonlinear diffusion equation, we establish that the mean waiting times of spreading obey a scaling law in dependence on energy. Moreover, we show that the spreading exponents very weakly depend on the level of disorder.
We investigate the OLS-based estimator s2 of the disturbance variance in the standard linear regression model with cross section data when the disturbances are homoskedastic, but spatially correlated. For the most popular model of spatially autoregressive disturbances, we show that s2 can be severely biased in finite samples, but is asymptotically unbiased and consistent for most types of spatial weighting matrices as sample size increases.