Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOperator limit of the circular beta ensemble
82
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We provide a precise coupling of the finite circular beta ensembles and their limit process via their operator representations. We prove explicit bounds on the distance of the operators and the corresponding point processes. We also prove an estimate on the beta-dependence in the $text{Sine}_{beta}$ process.
rate research
Read More
We prove an operator level limit for the circular Jacobi $beta$-ensemble. As a result, we characterize the counting function of the limit point process via coupled systems of stochastic differential equations. We also show that the normalized characteristic polynomials converge to a random analytic function, which we characterize via the joint distribution of its Taylor coefficients at zero and as the solution of a stochastic differential equation system. We also provide analogous results for the real orthogonal $beta$-ensemble.
We will prove the Berry-Esseen theorem for the number counting function of the circular $beta$-ensemble (C$beta$E), which will imply the central limit theorem for the number of points in arcs. We will prove the main result by estimating the characteristic functions of the Prufer phases and the number counting function, which will imply the the uniform upper and lower bounds of their variance. We also show that the similar results hold for the Sine$_beta$ process. As a direct application of the uniform variance bound, we can prove the normality of the linear statistics when the test function $f(theta)in W^{1,p}(S^1)$ for some $pin(1,+infty)$.
We prove rates of convergence for the circular law for the complex Ginibre ensemble. Specifically, we bound the expected $L_p$-Wasserstein distance between the empirical spectral measure of the normalized complex Ginibre ensemble and the uniform measure on the unit disc, both in expectation and almost surely. For $1 le p le 2$, the bounds are of the order $n^{-1/4}$, up to logarithmic factors.
The soft and hard edge scaling limits of $beta$-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators. It has been shown that by tuning the parameter of the hard edge process one can obtain the soft edge process as a scaling limit. We prove that this limit can be realized on the level of the corresponding random operators. More precisely, the random operators can be coupled in a way so that the scal
Various mixing properties of $beta$-, $beta$- and Gaussian Delaunay tessellations in $mathbb{R}^{d-1}$ are studied. It is shown that these tessellation models are absolutely regular, or $beta$-mixing. In the $beta$- and the Gaussian case exponential bounds for the absolute regularity coefficients are found. In the $beta$-case these coefficients show a polynomial decay only. In the background are new and strong concentration bounds on the radius of stabilization of the underlying construction. Using a general device for absolutely regular stationary random tessellations, central limit theorems for a number of geometric parameters of $beta$- and Gaussian Delaunay tessellations are established. This includes the number of $k$-dimensional faces and the $k$-volume of the $k$-sk
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
