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<p>We establish a direct correspondence between the Lanczos approach and the orthogonal polynomials approach in random matrix theory. In the large-<i>N</i> and continuum limits, the average Lanczos coefficients and the recursion coefficients become equivalent, with the precise mapping <i>b</i>(1 – <i>x</i>) = <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <msqrt> <mrow> <mi>R</mi> <mfenced close=")" open="("> <mi>x</mi> </mfenced> </mrow> </msqrt> </math></EquationSource> <EquationSource Format="TEX">\( \sqrt{R(x)} \)</EquationSource> </InlineEquation> and <i>a</i>(1 – <i>x</i>) = <i>S</i>(<i>x</i>). 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.</p>

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Lanczos meets orthogonal polynomials

  • 期刊论文

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  • 发表日期：

    2026年5月20日

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  • Le-Chen Qu

摘要

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) = R 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.