Extracting universal scaling functions of rigidity transitions from an effective medium theory

Rigidity transitions in random systems, such as jamming (J) and rigidity percolation (RP), have long evaded description by the usual framework of critical phenomena. The coherent potential approximation (CPA), a type of effective medium theory, has served in the past as a valuable tool to predict dynamics and transition points in randomly percolated lattices. We leverage the analytically tractable self-consistency equations for the self-energy in the CPA to express physically observable quantities, such as frequency-dependent viscoelastic moduli and correlation functions, in the usual scaling framework of critical phenomena [1]. We find the scaling behavior of these transitions in two spatial dimensions to be modified from that of higher dimensions – including a dangerous irrelevant variable that modifies the low-energy physics and logarithms that appear in the scaling functions [2].