The Hungarian physicist Eugene Wigner introduced random matrix models in
physics to describe the energy spectra of atomic nuclei. As such, the main goal
of Random Matrix Theory (RMT) has been to derive the eigenvalue statistics of
matrices drawn from a given distribution. 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. In the
resulting matrix, the diagonal and off-diagonal Lanczos coefficients control
transition amplitudes between elements of a distinguished basis of states. We
connect these two approaches to the quantum mechanics of complex systems by
deriving analytical formulae relating the potential defining a general RMT, or,
equivalently, its density of states, to the Lanczos coefficients and their
correlations. In particular, we derive an integral relation between the average
Lanczos coefficients and the density of states, and, for polynomial potentials,
algebraic equations that determine the Lanczos coefficients from the potential.
We obtain these results for generic initial states in the thermodynamic limit.
As an application, we compute the time-dependent “spread complexity” in
Thermo-Field Double states and the spectral form factor for Gaussian and
Non-Gaussian RMTs.

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