On Gaussian Comparison Inequality and Its Application to Spectral Analysis of Large Random Matrices

Recently, Chernozhukov, Chetverikov, and Kato [Ann. Statist. 42 (2014) 1564--1597] developed a new Gaussian comparison inequality for approximating the suprema of empirical processes. This paper exploits this technique to devise sharp inference on spectra of large random matrices. In particular, we show that two long-standing problems in random matrix theory can be solved: (i) simple bootstrap inference on sample eigenvalues when true eigenvalues are tied; (ii) conducting two-sample Roy's covariance test in high dimensions. To establish the asymptotic results, a generalized $\epsilon$-net argument regarding the matrix rescaled spectral norm and several new empirical process bounds are developed and of independent interest.Comment: to appear in Bernoull

Dynamics and correlation length scales of a glass-forming liquid in quiescent and sheared conditions

We numerically study dynamics and correlation length scales of a colloidal liquid in both quiescent and sheared conditions to further understand the origin of slow dynamics and dynamic heterogeneity in glass-forming systems. The simulation is performed in a weakly frustrated two-dimensional liquid, where locally preferred order is allowed to develop with increasing density. The four-point density correlations and bond-orientation correlations, which have been frequently used to capture dynamic and static length scales $\xi$ in a quiescent condition, can be readily extended to a system under steady shear in this case. In the absence of shear, we confirmed the previous findings that the dynamic slowing down accompanies the development of dynamic heterogeneity. The dynamic and static length scales increase with $\alpha$-relaxation time $\tau_{\alpha}$ as power-law $\xi\sim\tau_{\alpha}^{\mu}$ with $\mu>0$. In the presence of shear, both viscosity and $\tau_{\alpha}$ have power-law dependence on shear rate in the marked shear thinning regime. However, dependence of correlation lengths cannot be described by power laws in the same regime. Furthermore, the relation $\xi\sim\tau_{\alpha}^{\mu}$ between length scales and dynamics holds for not too strong shear where thermal fluctuations and external forces are both important in determining the properties of dense liquids. Thus, our results demonstrate a link between slow dynamics and structure in glass-forming liquids even under nonequilibrium conditions.Comment: 9 pages, 17 figures. Accepted by J. Phys.: Condens. Matte