Random multiplicative processes and the response functions of granular packings

It has recently been suggested that the property of isostaticity of the contact network of a frictionless polydisperse granular packing in the limit of low applied pressure is responsible for some of the anomalous static behaviour of packings. In this paper we discuss the fact that, on disordered isostatic networks, displacement-displacement and stress-stress static Green functions are described by coupled random multiplicative processes and thus have a truncated power-law distribution, with a cut-off that grows exponentially with distance. The expectation values of Green functions on these systems differ from observed averages by an exponentially large factor unless the number of samples over which averages are taken is exponentially large. Thus predicted averages will seldom be observed in experiments. If the external pressure is increased sufficiently, excess contacts are created, the packing becomes hyperstatic, and the above-mentioned anomalous properties disappear because Green functions now have a bounded distribution. Thus the low-pressure, isostatic, limit is a critical point where the Green function distribution becomes scale-free. This criticality is induced by multiplicative noise.