Gernot Akemann: Universal Local Statistics of Lyapunov exponents

Abstract: We consider the product of M complex Ginibre random matrices of sizes N &times N, being a simple toy model for chaotic dynamical systems.
While the behaviour of the Lyapunov exponents of the product matrix was known in the limit M to inifinity at finite N, taking deterministic values, recent progress has been made at finite M and N. The corresponding exponents follow a determinantal point process and thus all correlation functions of the squared singular values of the product matrix are known. This has allowed us to take a double scaling limit for N,M to infinity, where the Lyapunov exponents exhibits a rich variety of correlations, interpolating between deterministic behaviour and the sine- or Airy-kernel of a single random matrix, depending on the location in the spectrum. Surprisingly, in the bulk we find the same limiting interpolating kernel as for Dyson's Brownian motion for certain initial conditions as constructed by Johansson.

This is joint work with Zdzislaw Burda and Mario Kieburg
[arXiv:1809.05905 [math-ph]].