Random homomorphisms into the orthogonality graph

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 an appropriate sparsity condition on the subgraphs. These orthogonality graphs exhibit the main difficulties of defining subgraphs in the "middle" range, and so we expect their study to serve as a key example to defining subgraph densities in more general Markov spaces. Interest in studying homomorphisms of a finite graph G into orthogonality graphs is supported by the fact that such homomorphisms are just the orthonormal representations of the complementary graph.