Holo-Club: Lanczos Meets Orthogonal Polynomials

Title: Lanczos Meets Orthogonal Polynomials

Speaker: Le-Chen Qu

Venue & Time: 11:30 / Red Room

Abstract: In this talk, I will explain the connection between the Lanczos algorithm and orthogonal polynomials and its application to spread complexity in large-$N$ random matrix models. The three-term recurrence relation for orthogonal polynomials has the same structure as the Lanczos recursion, allowing the polynomial recursion coefficients to be identified with the Lanczos coefficients. For an infinite-temperature thermofield double state, equations for these coefficients follow from the Coulomb-gas equation. In one-cut matrix models, the Lanczos coefficients approach constants at large Krylov index, leading to linear growth of spread complexity at late times. A Bloch-wave analysis expresses the asymptotic growth rate as a spectral average of the group velocity. For even potentials, differences between consecutive squared Lanczos coefficients also admit a geometric interpretation as generating functions for planar maps whose marked legs are separated by a fixed geodesic distance. Quartic matrix models and the double-scaled SYK model illustrate these results. The talk concludes with the large-index asymptotics of recursion coefficients for general polynomial potentials and the gradient catastrophe that appears in the continuum limit of asymmetric models.