  Random matrix models provide a phenomenological description of a vast variety
of physical phenomena. Prominent examples include the eigenvalue statistics of
quantum (chaotic) systems, which are conveniently characterized using the
spectral form factor (SFF). Here, we calculate the SFF of unitary matrix
ensembles of infinite order with the weight function satisfying the assumptions
of Szeg\"o's limit theorem. We then consider a parameter-dependent critical
ensemble which has intermediate statistics characteristic of
ergodic-to-nonergodic transitions such as the Anderson localization transition.
This same ensemble is the matrix model of $U(N)$ Chern-Simons theory on $S^3$,
and the SFF of this ensemble is proportional to the HOMFLY invariant of
$(2n,2)$-torus links with one component in the fundamental and one in the
antifundamental representation. This is one of a large class of ensembles
arising from topological field and string theories which exhibit intermediate
statistics. Indeed, the absence of a local order parameter suggests that it is
natural to characterize ergodic-to-nonergodic transitions using topological
tools, such as we have done here.
