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Experimental results for covalent glasses have highlighted the existence of anew self-organized phase due to the tendency of glass networks to minimizeinternal stress. Recently, we have shown that an equilibrated self-organizedtwo-dimensional lattice-based model also possesses an intermediate phase inwhich a percolating rigid cluster exists with a probability between zero andone, depending on the average coordination of the network. In this paper, westudy the properties of this intermediate phase in more detail. We find thatmicroscopic perturbations, such as the addition or removal of a single bond,can affect the rigidity of macroscopic regions of the network, in particular,creating or destroying percolation. This, together with a power-lawdistribution of rigid cluster sizes, suggests that the system is maintained ina critical state on the rigid/floppy boundary throughout the intermediatephase, a behavior similar to self-organized criticality, but, remarkably, in athermodynamically equilibrated state. The distinction between percolating andnon-percolating networks appears physically meaningless, even though thepercolating cluster, when it exists, takes up a finite fraction of the network.We point out both similarities and differences between the intermediate phaseand the critical point of ordinary percolation models withoutself-organization. Our results are consistent with an interpretation of recentexperiments on the pressure dependence of Raman frequencies in chalcogenideglasses in terms of network homogeneity.Comment: 20 pages, 18 figure

Self-organized criticality in the intermediate phase of rigidity percolation