Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSemicircle law for generalized Curie-Weiss matrix ensembles at subcritical temperature
71
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Hochstattler, Kirsch, and Warzel showed that the semicircle law holds for generalized Curie-Weiss matrix ensembles at or above the critical temperature. We extend their result to the case of subcritical temperatures for which the correlations between the matrix entries are stronger. Nevertheless, one may use the concept of approximately uncorrelated ensembles that was first introduced in the paper mentioned above. In order to do so one needs to remove the average magnetization of the entries by an appropriate modification of the ensemble that turns out to be of rank 1 thus not changing the limiting spectral measure.
rate research
Read More
We derive and compare various forms of local semicircle laws for random matrices with exchangeable entries which exhibit correlations that decay at a very slow rate. In fact, any $l$-point correlation will decay at a rate of $N^{-l/2}$. We call our ensembles emph{of Curie-Weiss type}, and Curie-Weiss($beta$)-distributed entries are admissible as long as $betaleq 1$.
In the present paper, we give a simple proof of the level density of fixed trace square ensemble.We derive the integral equation of the level density of fixed trace square ensemble.Then we analyze the asymptotic behavior of the level density.
In this paper we present a criterion for the covering condition of the generalized random matrix ensemble, which enable us to verify the covering condition for the seven classes of generalized random matrix ensemble in an unified and simpler way.
Using the results on the $1/n$-expansion of the Verblunsky coefficients for a class of polynomials orthogonal on the unit circle with $n$ varying weight, we prove that the local eigenvalue statistic for unitary matrix models is independent of the form of the potential, determining the matrix model. Our proof is applicable to the case of four times differentiable potentials and of supports, consisting of one interval.
The celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off-diagonal pairs of elements, having applications to a wide range of physical and biological systems. Here, we investigate the generalization of this law to random matrices exhibiting higher-order cyclic correlations between $k$-tuples of matrix entries. We show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with $k$-fold rotational symmetry. This hypotrochoid law applies to full matrices as well as sparse ones, and thereby holds with remarkable universality. We further extend our analysis to matrices and graphs with competing cycle motifs, which are described more generally by polytrochoid spectral boundaries.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
