Rigorous solution for the elasticity of diluted Gaussian spring networks

We present a rigorous solution of the elasticity of the diluted Gaussian sp ring networks (DGSNs) at zero temperature. We show that the deformation of a diluted DGSN is homogeneous provided that the displacements of the partic les on the boundary are homogeneous. It follows that at zero temperature th e nonvanishing elastic stiffness coefficients are proportional to the hydro static pressure in both two and three dimensions. Follows a rigorous proof of the equivalence of the elasticity of the DGSN and the conductance of the random resistor network at zero temperature.