Lanczos meets orthogonal polynomials
Abstract
A bstract 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 ) = $$ \sqrt{R(x)} $$ 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.
Fields of Study & Subjects
Random Matrices and Applications
Matrix Theory and Algorithms
Mathematical functions and polynomials
Publication details
- Journal / source
- Journal of High Energy Physics
- Publisher
- Springer Science and Business Media LLC
- Volume
- 2026
- Issue
- 5
- Publication date
- 20 May 2026
- Language
- EN