Analytic continuation over complex landscapes*

Energy landscapes with a superextensive number of stationary points appear in models of structural and spin glasses, data science, strongly correlated electron systems, and string theory. In spin glasses, a lot is known about the structure of generic random landscapes made from real polynomials. Little is known about complex landscapes of complex variables, which appear explicitly in models of random lasers and quantum systems and implicitly in the analytic continuation of classical glasses. In the real case the statistics of stationary points by their energy and their number of unstable directions informs dynamics and geometry. In the complex case all stationary points are saddles with the same number of unstable directions, and whether their spectrum is gapped determines their local geometry. We describe what different landscape structures imply for analytic continuation.