Related papers: Lanczos Meets Orthogonal Polynomials

Lanczos Meets Orthogonal Polynomials

URL: http://arxiv.org/abs/2512.15857v1
Date: Wed, 17 Dec 2025 19:00:02 GMT
Title: Lanczos Meets Orthogonal Polynomials
Authors: Le-Chen Qu,
Abstract summary: In large-$N$ and continuum limits, the average Lanczos coefficients and the recursion coefficients become equivalent.<n>We show that the two formalisms yield identical expressions for the leading density states.
Score: 0.0
License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
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.

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