Using point-to-set correlations to probe unjamming of frictionless grains

We present a detailed analysis of the unjamming transition in 2D frictionless disk packings using a static correlation function that has been widely used to study disordered systems. We show that this point-to-set (PTS) correlation function exhibits a dominant length scale that diverges as the unjamming transition is approached through decompression. In addition, we identify deviations from mean-field predictions, and present detailed analysis of the origin of non-mean-field behavior. A mean-field bulk-surface argument is reviewed. Corrections to this argument are identified, which lead to a change in the functional form of the critical PTS boundary size R0. An entropic description of the origin of the correlations is presented, and simple rigidity assumptions are shown to predict the functional form of R0 as a function of the pressure P