  We explore the behavior of spatially heterogeneous elastic moduli as well as
the correlations between local moduli in model solids with short-range
repulsive potentials. We show through numerical simulations that local elastic
moduli exhibit long-range correlations, similar to correlations in the local
stresses. Specifically, the correlations in local shear moduli exhibit
anisotropic behavior at large lengthscales characterized by pinch-point
singularities in Fourier space, displaying a structural pattern akin to shear
stress correlations. Focussing on two-dimensional jammed solids approaching the
unjamming transition, we show that stress correlations exhibit universal
properties, characterized by a quadratic $p^2$ dependence of the correlations
as the pressure $p$ approaches zero, independent of the details of the model.
In contrast, the modulus correlations exhibit a power-law dependence with
different exponents depending on the specific interaction potential.
Furthermore, we illustrate that while affine responses lack long-range
correlations, the total modulus, which encompasses non-affine behavior,
exhibits long-range correlations.
