A diverging length scale in the structure of jammed systems

As temperature is lowered, motion becomes more sluggish.  Below the glass transition temperature, the dynamics of super-cooled liquids becomes so slow that the system falls out of equilibrium.  One hypothesis for this dynamic arrest is that it is due to a thermodynamic phase transition with a diverging length scale.  However, there is scant evidence of such a length scale appearing in the structure.  Motivated by this, we study another amorphous system that undergoes a phase transition: jammed soft repulsive spheres at zero temperature.  We have discovered a subtle correlation length, associated with the local coordination of particles, that is not seen in the two-point correlation function, g(r).  We argue that this scale plays an important role in determining the local rigidity of the system, and diverges with an exponent 2/(d+1) as the jamming transition is approached.