A Tale of Two Hungarians: Tridiagonalizing Random Matrices

TL;DR

This work builds a bridge between Random Matrix Theory and the Lanczos recursion method by deriving analytical relations that connect the average Lanczos coefficients to the density of states in generic RMTs. In the large-N limit, it yields an integral equation ρ(E) = ∫_0^1 dx [H(4 b(x)^2 − (E − a(x))^2)]/(π √{4 b(x)^2 − (E − a(x))^2}) that determines the Lanczos coefficients from the potential, with polynomial potentials leading to algebraic relations for a(x) and b(x). The authors develop saddle-point techniques to obtain one- and two-point functions of the Lanczos data, revealing when a(x) vanishes (even DOS) and when a covariance between a and b appears (non-Gaussian ensembles). They demonstrate that knowledge of the Lanczos statistics suffices to reproduce long-time observables like the spectral form factor and spread complexity for Thermo-Field Double states, establishing a practical link between static spectral data and dynamical complexity in chaotic 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.

Figures (5)

• Figure 1: Graph of the Lanczos coefficients $a(x),b(x)$ for one instance (light colors) and an average over $256$ instances (moderate colors) of size $N=1024$ random matrices with potentials $V_g$, $V_s$ and $V_q$ from left to right, along with the analytical solution for the average values (dark colors, continuous). The analytical solution overlaps well with the averaged values.
• Figure 2: Graph of the variance of $a(x),b(x)$ averaged over $256$ samples of $N=1024$ random matrices with potentials $V_g$, $V_s$ and $V_q$ from left to right (light colors), along with the analytical calculations of the variance (dark colors, continuous).
• Figure 3: SFF for $N=1024$ random matrices distributed according to the potentials $V_g$, $V_s$, $V_q$, averaged over $256$ samples (in orange), as well as the average computed from $256$ samples based on the one and two-point functions of the Lanczos coefficients (in blue).
• Figure 4: Spread complexity of the TFD state for $N=1024$ random matrices distributed according to the potentials $V_g$, $V_s$, $V_q$, averaged over $256$ samples (in orange), as well as the average computed from $256$ samples based on only the one and two-point functions of the Lanczos coefficients (in blue).
• Figure 5: Values of $C^n_m$ for various $m$ (columns) and $n$ (rows). Blank spots are zero. The first three columns appear in the online encyclopedia of integer sequences as OEIS A000302, A000346, A008549, see oeis.

Theorems & Definitions (6)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof