failure, on general grounds, we expect that elastic heterogeneities should play an equally important role. It would therefore be interesting to explore how the elastic heterogeneities discovered in the present work modify the macroscopic elasticity of rubbery materials. Such a program is left for a future work.

The outline of this paper is as follows. In Section \[SEC:RLPM\], we analyze the semi-microscopic RLPM using the tools of vulcanization theory. Specifically, we use the replica method [@Mezard1987] to study a model network consisting of randomly linked particles, which exhibits a continuous phase transition from the liquid state to the random solid state, paying particular attention to the Goldstone fluctuations of the random solid state, which, as we have mentioned above, are related to the elastic shear deformations of the random solid state. In Section \[SEC:Phen\], we propose a nonlocal phenomenological model of a random elastic medium, and subsequently derive it from the RLPM, by identify that this phenomenological model is the low energy theory (i.e., it captures the Goldstone fluctuations) of the RLPM in the random solid state. Through this correspondence we learn information of the statistics of the quenched random nonlocal kernel. In Section \[SEC:relaxation\], we study the relaxation of 