  Substrates, and layered media in general, are ubiquitous, affect the
properties of whatever is in their vicinity, and their influence is, in an
arbitrary framework, challenging to quantify analytically, especially for large
arrays which escape explicit numerical treatment due to the computational
burden. In this work, we develop a versatile T-matrix based framework in which
we generalize the coupled multipole model towards arbitrarily high multipole
orders and substrate-supported arrays. It allows us to study
substrate-supported random/amorphous arrays of high index dielectric
nanoparticles which are of wide interest due to relatively low losses and a
highly tunable optical response, making them promising elements for
nanophotonic devices. We discuss how multipole coupling rules evolve in the
presence of a substrate in amorphous arrays for three interaction mechanisms:
direct coupling between particles, substrate-mediated interparticle coupling
and substrate-mediated self-coupling. We show the interplay between array
density, distance from the substrate and its refractive in determining the
optical response of an array. As an example, we use this framework to analyze
refractometric sensing with substrate-supported arrays and demonstrate that the
substrate plays a crucial role in determining the array sensitivity.
