Universal scaling of disordered rigidity transitions: RG flows near two dimensions

Cells and tissues can often be situated near in phase space to a transition in their rigidity, where their elastic moduli vanish. We investigate the properties of a model that makes predictions for effective viscoelastic moduli of driven systems near this transition. By casting the self-consistent solutions of the model into a scaling form, we find anomalous logarithmic corrections that appear in two dimensions, which have been previously detected in simulations of diluted elastic networks. We put forth a detailed description of these logarithmic corrections by investigating the topology of the renormalization group flows in different spatial dimensions, showing that two dimensions is the upper critical dimension predicted by the theory. We give analytic formulas for the values of critical exponents and shapes of scaling collapse plots for all linear response properties above, in, and below the upper critical dimension. These predictions could have relevance for experimental measurements of two-dimensional disordered rigid membranes near their rigid-to-floppy transition.