arXiv subjects

Mengchun Cai

Publications and source records attributed to Mengchun Cai.

3 recordsLinked to original sources

arXiv2026

Mesoscopic Rates of Convergence for Complex Wishart Matrices at the Leftmost Spectrum Edge

This paper establishes mesoscopic rates of convergence in the $L^1$-Wasserstein distance for eigenvalue determinantal point processes (DPPs) derived from the Laguerre Unitary Ensemble (LUE) to the corresponding limiting point process (Airy process) as the dimension goes to infinity. Specifically, we prove convergence rates at the leftmost edge of the LUE spectrum, which corresponds to the least eigenvalue. These results are termed mesoscopic because they allow for a direct comparison of point counts between the convergent DPPs and their limits over a range of scales.

math.PR↗

arXiv2025

Rates of Bulk Convergence for Ensembles of Classical Compact Groups

This paper considers random matrices distributed according to Haar measure in different classical compact groups. Utilizing the determinantal point structures of their nontrivial eigenangles, with respect to the $L_1$-Wasserstein distance, we obtain the rate of convergence for different ensembles to the sine point process when the dimension of matrices $N$ is sufficiently large. Specifically, the rate is roughly of order $N^{-2}$ on the unitary group and of order $N^{-1}$ on the orthogonal group and the compact symplectic group.

math.PR↗

arXiv2025

Mesoscopic Rates of Convergence for Hermitian Unitary Ensembles

This paper provides mesoscopic rates of convergence (ROC) with respect to the $L^1$-Wasserstein distance for the eigenvalue determinantal point processes (DPPs) from the three major Hermitian unitary ensembles, the Gaussian Unitary Ensemble (GUE), the Laguerre Unitary Ensemble (LUE), and the Jacobi Unitary Ensemble (JUE) to their limiting point processes. We prove ROCs for the bulk of the GUE spectrum, the hard edge of the LUE spectrum, and the soft edges of the GUE, LUE, and JUE spectrums. These results are called mesoscopic because we are able to directly compare the point counts between the converging and limit DPPs in a range of scales. We are able to achieve these results by controlling the trace class norm of the integral operators determined by the DPP kernels.

math.PR↗