Can a large jammed packing be assembled from smaller ones ?

The principle of equivalence of ensembles asserts that, in the thermodynamic limit, a system with periodic boundary conditions behaves identically to a subsystem of the same size, cut out from a larger system. We compare these two ensembles on finite length scales in the case of amorphous jammed packings of soft spheres at zero temperature. Focusing on the statistics of the contact fluctuations, we find that systems with periodic boundary conditions have significantly smaller fluctuations compared to the subsystems. This difference is largest near the jamming transition. Moreover, these two ensembles converge only at a surprisingly large system size. The crossover to the thermodynamic limit defines a length scale for each ensemble. Surprisingly, these diverge, as a function of the distance to the transition, with two different exponents. We argue that this disparity is the result of the system being above the upper critical dimension and that, based on the values of the exponents, the upper critical dimension can be measured.