Anomalous force diffusion in nearly ordered packings of frictionless discs

We derive analytic expressions for force propagation in packings of frictionless discs with a narrow distribution of disc sizes, by expanding to first order about the known ordered solution. The distribution of contact forces P(f) is found to be narrow at the upper surface, and broadens at a rate that varies with depth, being superdiffusive near the surface until crossing over to a subdiffusive regime near the fixed base. Furthermore, the response to an isolated load propagates along the edge of a 'cone', as in the ordered case, but fluctuates under ensemble, averaging by an amount that depends purely on height, not on the lateral position. Finally, we comment on ways in which the analytical framework presented here could be extended to a wider range of granular packings.