Real-space renormalization of randomly vacated lattices: a renormalization group for jamming?

Jamming occurs in granular materials, as well as in emulsions, dense suspensions, and other amorphous, particulate systems. When the pack- ing 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 that diverge near φ = φ_c, but there has been no explicit spatial renormalization group (RG) scheme that has captured this transition. Here, I present a candidate for such a scheme, using a block-spin-like transformation of a randomly vacated lattice of grains. I define a real-space RG transformation based on local mechanical stability. This model displays a critical packing fraction φ_c and gives estimates of critical exponents in two and three dimensions.