Lanczos Meets Orthogonal Polynomials

TL;DR

This work establishes a direct mapping between the Lanczos approach to quantum dynamics and the orthogonal-polynomials framework in random matrix theory, showing that in the large-$N$ continuum limit the average Lanczos coefficients $(a,b)$ coincide with the recursion data $(R,S)$ via $\sqrt{R(x)} = b(1-x)$ and $S(x) = a(1-x)$. This equivalence yields identical leading densities of states and equips orthogonal polynomials with a Krylov-polynomial interpretation, enabling a unified Krylov-dynamics treatment. The Gaussian Unitary Ensemble is worked out analytically to illustrate the correspondence: $\langle a_n\rangle=0$, $\langle b_n^2\rangle=1-n/N$, $S_n=0$, $R_n=n/N$, with the semicircle law recovered and explicit Krylov amplitudes and spread complexity derived. The results pave the way for applying either formalism interchangeably to spectral questions and hint at deeper connections to emergent geometry and gravity in related quantum-chaotic systems.

Abstract

We establish a direct correspondence between the Lanczos approach and the orthogonal polynomials approach in random matrix theory. In the large-$N$ and continuum limits, the average Lanczos coefficients and the recursion coefficients become equivalent, with the precise mapping $\sqrt{R(x)} = b(1-x)$ and $S(x) = a(1-x)$. As a result, the two formalisms yield identical expressions for the leading density of states. We further analyze the Krylov dynamics associated with the recursion coefficients and show that the orthogonal polynomials admit a natural interpretation as Krylov polynomials. This picture is realized explicitly in the Gaussian Unitary Ensemble, where all quantities can be computed analytically.