Statistical features of quantum chaos using the Krylov operator complexity

We study the statistical properties of Lanczos coefficients over an ensemble of random initial operators generating the Krylov space. We propose two statistical quantities that are important in characterizing the complexity: the average correlation matrix $\langle x_{i} x_{j}\rangle$ of Lanczos coefficients and the resulting distribution of the variance of Lanczos coefficients. Their resulting statistics are the Wishart distribution and the (rescaled) chi-square distribution respectively, which are independent of the distributions of initial operators and become the normal distribution in the case of large matrix size. As a numerical example, we use the typical billiard system with an integrability-breaking term and choose samples of random initial operators from given probability distributions (GOE, GUE and the uniform distribution). It agrees with the phenomenological analysis and further interesting behaviors are obtained, which indicates a consistent connection between RMT, Anderson localization and Krylov complexity.