Simple argument for emergent anisotropic stress correlations in disordered solids

It is now well-established that mechanical equilibrium in athermal disordered solids gives rise to anisotropic spatial correlations of the coarse-grained stress field^1–9 that decay in space as $1/rđ$⁠, where r is the distance from the origin and $đ$ denotes the spatial dimension. In this Note, we present a simple, geometry based argument for the scaling form of the emergent spatial correlations of the stress field in disordered solids. The presented approach bears some conceptual similarities with the field-theoretic approach of Refs. 5–7.

Consider a disordered solid whose constituent particles interact via pairwise, radially symmetric interactions. We denote particle coordinates by x_i, the radius vector between a pair i, j of interacting particles by x_ij ≡ x_j − x_i, and the distance between them by $rij≡xij⋅xij$⁠...