A tale of two hungarians: tridiagonalizing random matrices
The Hungarian physicist Eugene Wigner introduced random matrix models in physics to describe the energy spectra of atomic nuclei. The Wigner approach gives powerful insights into the properties of complex, chaotic systems in thermal equilibrium. Another Hungarian, Cornelius Lanczos, suggested a method of reducing the dynamics of any quantum system to a one-dimensional chain by tridiagonalizing the Hamiltonian relative to a given initial state. This method suggests a computable notion of complexity which we describe in detail. We then connect the Wigner/Lanczos approaches by analyzing RMT in time-dependent scenarios. This is first accomplished by numerically studying time-evolved thermofield double states in chaotic systems, where our complexity measure is connected to the Spectral Form Factor in RMT, showing parallel regimes. Secondly, to approach these problems analytically, we initiate a novel approach to Random Matrix Theory based on matrix tridiagonzalition, deriving the statistics of the tridiagonal matrix elements.