Subgraph densities have been defined, and served as basic tools, both in the case
of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree
graph sequences). While limit objects have been described for the "middle ranges",
the notion of subgraph densities in these limit objects remains elusive. We define
subgraph densities in the orthogonality graphs on the unit spheres in dimension d,
under appropriate sparsity condition on the subgraphs. These orthogonality graphs
exhibit the main difficulties of defining subgraphs the "middle" range, and so we
expect their study to serve as a key example to defining subgraph densities in more
general Markov spaces.
The problem can also be formulated as defining and computing random orthogonal representations
of graphs. Orthogonal representations have played a role in information theory, optimization,
rigidity theory and quantum physics, so to study random ones may be of interest from
the point of view of these applications as well.
