  We investigate the emergence of isotropic linear elasticity in amorphous and
polycrystalline solids, via extensive numerical simulations. We show that the
elastic properties are correlated over a finite length scale $\xi_E$, so that
the central limit theorem dictates the emergence of continuum linear isotropic
elasticity on increasing the specimen size. The stiffness matrix of systems of
finite size $L > \xi_E$ is obtained adding to that predicted by linear
isotropic elasticity a random one of spectral norm $(L/\xi_E)^{-3/2}$, in three
spatial dimensions. We further demonstrate that the elastic length scale
corresponds to that of structural correlations, which in polycrystals reflect
the typical size of the grain boundaries and length scales characterizing
correlations in the stress field. We finally demonstrate that the elastic
length scale affects the decay of the anisotropic long-ranged correlations of
locally defined shear modulus and shear stress.
