Recursion Coefficients and Krylov Dynamics in Polynomial Random Matrix Models

We study the recursion coefficients of orthogonal polynomials and their associated Krylov dynamics in random matrix models with high-degree and possibly asymmetric polynomial potentials. We develop a moment recursion method that, when combined with the recursive algorithm, provides an efficient construction of the recursion coefficients. We also obtain their large-n asymptotic behavior for general asymmetric potentials; for Nw_d=1, the leading asymptotic form of R_n reproduces Freud's conjecture. We apply this framework to an asymmetric quartic potential and to the double-scaled Sachdev-Ye-Kitaev (DSSYK) model. In both models, the recursion functions capture the overall qualitative behavior of the recursion coefficients, and the gradient catastrophes of the recursion functions are associated with "chaotic" transition regions in the recursion coefficients. For the quartic potential, such regions can occur in both R_n and S_n, whereas the DSSYK model can exhibit multiple transition regions in R_n, with the recursion function remaining accurate in the smooth intervals between them. Finally, we compute the corresponding spread complexity and find that transition regions do not qualitatively modify its behavior, while a two branch structure produces early time oscillations followed by monotonic growth.