a statistical characterization of the heterogeneous elasticity of soft random solids by starting from a semi-microscopic model and applying a body of techniques that we shall call vulcanization theory to it. In particular, we aim to obtain the mean values and disorder correlators of elastic parameters, such as the Lamé coefficients and the residual stress, in terms of the parameters of the semi-microscopic model, such as the density of cross-links, the excluded-volume interactions, etc. One of our key findings is that the disorder correlator of the residual stress is long ranged, as are all cross-disorder correlators between the residual stress and the Lamé coefficients. We also find that these disorder correlators are controlled by a universal scale parameter—independent of the microscopic details—that, moreover, controls the scale of the mean shear modulus. In addition, we characterize the nonaffininity of the deformations in terms of these parameters.

The strategy we adopt for accomplishing our goals involves a handshaking between two different analytical schemes. [^2] The first scheme follows a well-trodden path. We begin with a semi-microscopic model, the Randomly Linked Particle Model (RLPM) [@Mao2007; @Ulrich2006; @Broderix2002; @Mao2005], involving particle coordinates and quenched random interactions between them that represent the randomness that is 