Recursion Coefficients and Krylov Dynamics in Polynomial Random Matrix Models

This paper investigates the recursion coefficients of orthogonal polynomials in random matrix models with high-degree, asymmetric potentials. By identifying these coefficients with Lanczos coefficients in Krylov dynamics, the authors aim to characterize the system's behavior beyond the standard symmetric cases. They introduce a moment recursion method that avoids the computational complexity of traditional discrete string equations, allowing for the analysis of high-degree potentials. They validate this approach by applying it to an asymmetric quartic potential and the double-scaled Sachdev-Ye-Kitaev (DSSYK) model.

The study establishes that the large-n asymptotic behavior of the recursion coefficients for general asymmetric potentials reproduces Freud's conjecture. A key finding is the identification of gradient catastrophes in the continuum recursion functions, which signal the emergence of chaotic transition regions in the discrete recursion coefficients. While these transition regions appear in both the quartic and DSSYK models, the authors find that the recursion functions remain accurate in the smooth intervals between these regions. Furthermore, the analysis of spread complexity reveals that these chaotic transition regions do not qualitatively alter the growth of complexity, whereas a two-branch structure in the recursion coefficients induces early-time oscillations before settling into monotonic growth.

Understanding the behavior of recursion coefficients is essential for mapping random matrix theory to quantum gravity and string theory. By providing a robust numerical framework for asymmetric potentials and clarifying the relationship between gradient catastrophes and chaotic dynamics, this work extends the applicability of Krylov methods to a broader class of physical models. The results offer a clearer picture of how complexity evolves in systems with non-trivial potential landscapes, bridging the gap between mathematical orthogonal polynomial theory and physical quantum dynamics.