Abstract: The primordial perturbations inferred from observations of the Cosmic Microwave Background and the large-scale structure of the universe are quite simple, to a good approximation Gaussian and approximately scale-invariant, and consistent with some of the simplest models of single-field slow-roll inflation. Embeddings of inflation in string theory however tend to be rather complex, involving many fields (moduli and axions) with complicated interactions. Unfortunately, constructing explicit models of inflation with many fields is computationally very challenging, and not much is known about the observational predictions of complex many-field models of inflation.  In this talk, I will report on some recent advances on this problem: using techniques from non-equilibrium random matrix theory, ensembles of complicated inflationary models with many fields can be constructed, and their observational predictions computed. Strikingly, as the number of interacting fields (and hence the complexity of the model) increases, the power spectra of curvature perturbations simplify, and become more predictive. I will explain how these results can be understood in terms of universal properties of large, random matrices.