Jamming versus caging in three dimensional jamming percolation

We investigate a three-dimensional kinetically-constrained model that exhibits two types of phase transitions at different densities. At the jamming density ρJ there is a mixed-order phase transition in which a finite fraction of the particles become frozen, but the other particles may still diffuse throughout the system. At the caging density ρC > ρJ, the mobile particles are trapped in finite cages and no longer diffuse. The caging transition occurs due to a percolation transition of the unfrozen sites, and we numerically find that it is a continuous transition with the same critical exponents as random percolation.