We give large deviation results for random projections of $\ell^p$ balls. They quantify the well-know statement that two independently drawn vectors whose law is uniform on a high-dimensional sphere, are nearly orthogonal. We give both quenched large deviation principles (fixing the sequence of projection directions) and annealed large deviation principles (averaging over the sequence of projection directions). There is an analogy with random environments, and we present a variational formula relating the two rate functions.

The talk is based on joint work with Steven Soojin Kim and Kavita Ramanan, Brown University.