AI 中文总结
该研究针对含高阶非对称多项式势的随机矩阵模型，开发矩递归方法求解递归系数，应用于非对称四次势和DSSYK模型，分析其渐近行为、转变区域及传播复杂度特性。
AI 中文摘要
我们研究了具有高阶且可能非对称多项式势的随机矩阵模型中正交多项式的递归系数及其关联的Krylov动力学。我们开发了一种矩递归方法，该方法与递归算法结合可高效构建递归系数。我们还得到了一般非对称势下递归系数的大n渐近行为；当Nw_d=1时，R_n的主导渐近形式重现了弗洛伊德猜想。我们将该框架应用于非对称四次势和双标Sachdev-Ye-Kitaev（DSSYK）模型。在这两个模型中，递归函数捕捉了递归系数的整体定性行为，且递归函数的梯度灾变与递归系数中的“混沌”转变区域相关。对于四次势，这类区域可出现在R_n和S_n中，而DSSYK模型可在R_n中表现出多个转变区域，递归函数在这些区域间的平滑区间内仍保持准确。最后，我们计算了对应的传播复杂度，发现转变区域不会定性改变其行为，而双分支结构会产生早期振荡，随后呈单调增长。
英文摘要
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.
Commentsv1: 16 pages, 7 figures; v2: references added
