the phenomenological model to a stable state for any fixed randomness (i.e., any realization of disorder), due to random stresses and attractive interactions. We re-expand the free energy about this relaxed state to obtain the true elastic theory. This relaxed reference state is, however, still randomly stressed [@Alexander1998]; nevertheless, the stress in this state—the so-called *residual stress*—does satisfy the condition of mechanical equilibrium, viz., $\partial_i \Stre_{ij}(x)=0$. In its local limit, the proposed phenomenological model reproduces a version of Lagrangian continuum elasticity theory that features random Lamé coefficients and residual stresses. In this section, we also use the phenomenological model to explore the related issue of elastic heterogeneity, viz., the nonaffine way in which the medium responds to external stress. In Section \[SEC:Heterogeneity\], we arrive at predictions for the statistics of the quenched random elastic parameters that feature in the phenomenological model in the relaxed state, along with the statistics of nonaffine deformations. Thus we provide a first principles account of the heterogeneous elasticity of soft random solids. We conclude, in Section \[SEC:ConcDisc\], with a brief summary and discussion of our results.

Semi-microscopic approach: The randomly linked particle model {#SEC:RLPM}
=============================================================

Randomly linked particle model {#SEC:RLPMBasics}
------------------------------

The Randomly Linked Particle 