arXiv subjects
Publications and source records attributed to Trinh Khanh Duy.
The paper studies the limiting behavior of spectral measures of random Jacobi matrices of Gaussian, Wishart and MANOVA beta ensembles. We show that the spectral measures converge weakly to a limit distribution which is the semicircle distribution, Marchenko-Pastur distributions or Kesten-Mckey distributions, respectively. The Gaussian fluctuation around the limit is then investigated.
We study the limiting behavior of Gaussian beta ensembles in the regime where $βn = const$ as $n \to \infty$. The results are (1) Gaussian fluctuations for linear statistics of the eigenvalues, and (2) Poisson convergence of the bulk statistics. (2) is an alternative proof of the result by F.~Benaych-Georges and S.~Péché (2015) with the explicit form of the intensity measure.
The persistent homology of a stationary point process on ${\bf R}^N$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the limiting persistence diagram to have the full support. We also discuss a central limit theorem for persistent Betti numbers.
The convergence of the expectations of Betti numbers of Čech complexes built on binomial point processes in the thermodynamic regime is established.
The determinant of the Gaussian unitary ensemble matrix is show to be distributed as a product of independent chi random variables with parameters $1,3,3,5,5,\dots.$
An explicit formula for the mean spectral measure of a random Jacobi matrix is derived. The matrix may be regarded as the limit of Gaussian beta ensemble (G$β$E) matrices as the matrix size $N$ tends to infinity with the constraint that $N β$ is a constant.
Spectral measures of Wigner matrices are investigated. The Wigner semicircle law for spectral measures is proved. Regard this as the law of large number, the central limit theorem for moments spectral measure is also derived. The proof is based on moment method and combinatorial method.