theory of percolation [@Stauffer1994]. However, to study the [*elasticity*]{} that emerges at the gelation/vulcanization transition, and especially its heterogeneity, one needs a theory that incorporates not only the geometrical aspects, but also the equilibrium thermal fluctuations of the particle positions and the strong, qualitative changes that they undergo at the gelation/vulcanization transition. In the setting of rubber elasticity, although the classical theory is successful in explaining the entropic nature of the shear rigidity of rubber, it is essentially based on a single-chain picture and, as such, is incapable of describing the consequences of the long scale random structure of rubbery media, e.g., the random spatial variations in their local elastic parameters and the resulting nonaffinity of their local strain-response to macroscopic applied stresses.

The general problem of heterogeneous elasticity and nonaffine deformations has been studied in the setting of flexible polymer networks [@Rubinstein1997; @Glatting1995; @Holzl1997; @Svaneborg2005], and also semi-flexible polymer networks [@Head2003; @Heussinger2007], glasses [@Wittmer2002], and granular materials [@Utter2008]. Particularly noteworthy is the recent investigation by DiDonna and Lubensky of the general relationship between the spatial correlations of the nonaffine deformations and those of the underlying quenched random elastic parameters [@DiDonna2005].

The mission of the present work is to develop 