Lanczos Meets Orthogonal Polynomials

TL;DR

This paper establishes a direct link between the Lanczos algorithm and orthogonal polynomials in random matrix theory, showing their equivalence in the large-$N$ limit and providing explicit analysis for the Gaussian Unitary Ensemble.

Contribution

It reveals a precise correspondence between Lanczos coefficients and orthogonal polynomial recursion coefficients, unifying two formalisms in random matrix theory.

Findings

Lanczos and orthogonal polynomial approaches are equivalent in the large-$N$ limit.

Explicit mapping between Lanczos coefficients and polynomial recursion coefficients.

Analytical results obtained for the Gaussian Unitary Ensemble.

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 $b(1-x)=\sqrt{R(x)}$ and $a(1-x)=S(x)$. 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.