the relaxed state, but with coefficients that are explicit functions of the quenched random kernel. Thus, using the information about the statistics of the quenched random kernel obtained via the comparison with the RLPM, we are able to infer statistical information about the elastic properties of the random elastic medium in the relaxed state, which is of experimental relevance.

Why is it legitimate to identify the Goldstone theory arising from the microscopic model with the replica theory of the phenomenological model? The reason is that, within the schemes that we have chosen to analyze them, both models describe shear deformations not of the equilibrium state of the system but, rather, of the system immediately after cross-linking has been done but before any cross-linking-induced relaxation has been allowed to occur. [[ The equivalence between these two schemes is not based solely on the equality of free energies of the two models; it is also based on the identity of the physical meaning of the (replicated) deformation fields in the two theories, and thus the way that these fields couple to externally applied forces. Both schemes are descriptions of the elasticity of soft random solids displaying heterogeneous elastic properties, one from a 