Mesoscopic Statistics Research Articles

Dyson Brownian motion for general $$\beta $$ and potential at the edge

In this paper, we compare the solutions of the Dyson Brownian motion for general $$\beta $$ and potential V and the associated McKean–Vlasov equation near the edge. Under suitable conditions on the initial data and the potential V, we obtain optimal rigidity estimates of particle locations near the edge after a short time $$t={{\,\mathrm{o}\,}}(1)$$ . Our argument uses the method of characteristics along with a careful estimate involving an equation of the edge. With the rigidity estimates as an input, we prove a central limit theorem for mesoscopic statistics near the edge, which, as far as we know, has been done for the first time in this paper. Additionally, combining our results with Landon and Yau (Edge statistics of Dyson Brownian motion. arXiv:1712.03881 , 2017), we give a proof of the local ergodicity of the Dyson Brownian motion for general $$\beta $$ and potential at the edge, i.e., we show the distribution of extreme particles converges to the Tracy–Widom $$\beta $$ distribution in a short time.

Incomplete Determinantal Processes: From Random Matrix to Poisson Statistics

We study linear statistics of a class of determinantal processes which interpolate between Poisson and GUE/Ginibre statistics in dimension 1 or 2. These processes are obtained by performing an independent Bernoulli percolation on the particle configuration of of a log-gas confined in a general potential. We show that, depending on the expected number of deleted particles, there is a universal transition for mesoscopic statistics. Namely, at small scales, the point process still behaves according to random matrix theory, while, at large scales, it needs to be renormalized because the variance of any linear statistic diverges. The crossover is explicitly characterized as the superposition of a H^{1/2}- or H^1- correlated Gaussian noise and an independent Poisson process. The proof consists in computing the limits of the cumulants of linear statistics using the asymptotics of the correlation kernel of the process.

Fixed Energy Universality for Generalized Wigner Matrices

AbstractWe prove the Wigner‐Dyson‐Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.© 2016 Wiley Periodicals, Inc.

Mesoscopic simulations of phase distribution effects on the effective thermal conductivity of microgranular porous media

This paper analyzes the phase distribution effects on the effective thermal conductivity (ETC) of multi-phase microgranular porous media using mesoscopic statistics based numerical methods. A multi-parameter random generation-growth method, quartet structure generation set (QSGS), is developed for replicating microstructures of multi-phase granular porous media based on the macroscopic statistical information, such as the volume fractions and the phase interactions. The phase distribution characteristics and the interphase connections are controlled by adjusting the related parameters. Then the energy transport equations through porous media are solved by a lattice Boltzmann method developed by us with multi-phase conjugate heat transfer considered. The results indicate that a smaller average particle size could lead to a larger effective thermal conductivity of two-phase porous media for a certain porosity. For the anisotropic media, if the larger directional growth probability is along the direction of temperature gradient, the effective thermal conductivity in the parallel direction is enhanced as a result, and that in the vertical direction will be weakened. For multi-phase porous media, the degree of phase conglomeration is determined by the phase interactions. A larger liquid–liquid interaction leads to a higher degree of liquid phase conglomeration and therefore a larger effective thermal conductivity of the porous media.