Effective-medium theory for elastic systems with correlated disorder

Effective-medium approaches, such as the coherent potential approximation, are commonly used to describe the rigidity transition of disordered elastic networks. While successful in capturing the universal critical behavior of these systems near the onset of rigidity, its traditional formulation neglects spatial correlations in the underlying disorder. Here, we propose a multipurpose generalization of the coherent potential approximation to describe the mechanical behavior of elastic networks with spatially correlated disorder. We apply our theory to a simple rigidity-percolation model for colloidal gels and study the effects of correlations on both the critical point and the overall scaling behavior. We find that although the presence of spatial correlations (mimicking attractive interactions in gels) shifts the critical packing fraction to lower values, suggesting subisostatic behavior, the critical coordination number of the associated network remains isostatic. More importantly, we discuss how our theory can be employed to describe a large variety of systems with spatially correlated disorder.