发表机构
Instituto de Física Teórica UAM/CSIC; Departamento de Física Teórica, Universidad Autónoma de Madrid（UAM/CSIC理论物理研究所; 马德里自治大学理论物理系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在大N随机矩阵理论中，通过库仑气方程推导Lanczos方程，将Lanczos系数解释为平面地图生成函数，阐释了传播复杂度的增长规律及相关模型的应用。
AI 中文摘要
我们研究了大N随机矩阵理论中无限温度热场双态的传播复杂度，并探究其Lanczos系数的物理意义。主导态密度定义了一族正交多项式，其递推系数与相关Krylov动力学的Lanczos系数重合。我们从库仑气方程出发，推导出这组系数的一对非线性关系，将其命名为Lanczos方程。由于Lanczos系数在大Krylov指标下趋近于常数，传播复杂度在后期呈线性增长。我们进一步证明，Lanczos系数可作为平面地图的生成函数；对于偶势，差值b_{n+1}^2 - b_n^2生成带有两个支腿的地图，其标记支腿间的精确测地距离为n，因此传播复杂度加速算子等于两倍测地生成算子。我们在四次矩阵模型中阐释这些结果，其中四次耦合的展开计数四价平面地图；在双标度Sachdev–Ye–Kitaev模型中，辅助希尔伯特空间可等同于物理弦希尔伯特空间。
英文摘要
We study the spread complexity of the infinite-temperature thermofield double state in large-$N$ random matrix theory and investigate the physical meaning of its Lanczos coefficients $b_n$. The leading density of states defines a family of orthogonal polynomials whose recursion coefficients coincide with the Lanczos coefficients of the associated Krylov dynamics. Starting from the Coulomb-gas equation, we derive a pair of nonlinear relations for these coefficients, which we call the Lanczos equations. For normalizable one-cut ensembles, the Lanczos coefficients approach constants at large Krylov index, rendering the asymptotic Krylov chain translationally invariant and the late-time growth of spread complexity linear, with the growth rate given by the spectral average of the group velocity obtained from a Bloch-wave analysis. For an even potential, we further show that the difference between consecutive squared Lanczos coefficients, $b_{n+1}^2-b_n^2$, is the generating function for two-legged maps whose marked legs are separated by the exact geodesic distance $n$. Consequently, the spread complexity acceleration operator equals twice the geodesic generating operator. We illustrate these results in the quartic matrix model, where the expansion in the quartic coupling counts tetravalent planar maps, and in the double-scaled Sachdev--Ye--Kitaev model, where the auxiliary Hilbert space can be identified with the physical chord Hilbert space.
Commentsv3: 23 pages, 4 figures
