A real-space renormalization group for jamming

Jamming occurs in grains, emulsions, dense suspensions, and other amorphous, particle-based systems. As the packing fraction φ, defined as the ratio of particle volume to system volume, is increased past a critical value φ_c, a liquid-solid phase transition occurs, and grains are no longer able to rearrange. Previous studies have shown evidence of spatial correlations, corresponding to collective rearrangements, that diverge near φ ≈ φ_c, but there has been no explicit spatial renormalization group scheme that has captured this transition. Here, I present a candidate for such a scheme, using a simple block-spin-like transformation of a randomly vacated lattice of grains. I define a real-space renormalization transformation that coarse-grains the lattice based on local mechanical stability. This model displays a critical packing fraction φ_c and estimates of critical exponents in 2D and 3D that are consistent with previous work.