A Tale of Two Hungarians: Tridiagonalizing Random Matrices

TL;DR

This paper links random matrix theory and Lanczos tridiagonalization by deriving formulas that relate the potential of a matrix ensemble to its Lanczos coefficients, with applications to quantum complexity and spectral analysis.

Contribution

It introduces analytical relations connecting the potential defining a random matrix ensemble to its Lanczos coefficients, bridging RMT and quantum dynamics.

Findings

Derived an integral relation between Lanczos coefficients and the density of states.

Established algebraic equations for Lanczos coefficients for polynomial potentials.

Applied results to compute complexity and spectral form factors in quantum systems.

Abstract

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.