Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA Local Law for Singular Values from Diophantine Equations
77
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We introduce the $Ntimes N$ random matrices $$ X_{j,k}=expleft(2pi i sum_{q=1}^d omega_{j,q} k^qright) quad text{with } {omega_{j,q}}_{substack{1leq jleq N 1leq qleq d}} text{ i.i.d. random variables}, $$ and $d$ a fixed integer. We prove that the distribution of their singular values converges to the local Marchenko-Pastur law at scales $N^{-theta_d}$ for an explicit, small $theta_d>0$, as long as $dgeq 18$. To our knowledge, this is the first instance of a random matrix ensemble that is explicitly defined in terms of only $O(N)$ random variables exhibiting a universal local spectral law. Our main technical contribution is to derive concentration bounds for the Stieltjes transform that simultaneously take into account stochastic and oscillatory cancellations. Important ingredients in our proof are strong estimates on the number of solutions to Diophantine equations (in the form of Vinogradovs main conjecture recently proved by Bourgain-Demeter-Guth) and a pigeonhole argument that combines the Ward identity with an algebraic uniqueness condition for Diophantine equations derived from the Newton-Girard identities.
rate research
Read More
In the first part of this article, we proved a local version of the circular law up to the finest scale $N^{-1/2+ e}$ for non-Hermitian random matrices at any point $z in C$ with $||z| - 1| > c $ for any $c>0$ independent of the size of the matrix. Under the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case $ |z|-1=oo(1)$. Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge $ |z|-1=oo(1)$ up to scale $N^{-1/4+ e}$.
The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by a Haar orthogonal matrix.
We solve a class of BSDE with a power function $f(y) = y^q$, $q > 1$, driving its drift and with the terminal boundary condition $ xi = infty cdot mathbf{1}_{B(m,r)^c}$ (for which $q > 2$ is assumed) or $ xi = infty cdot mathbf{1}_{B(m,r)}$, where $B(m,r)$ is the ball in the path space $C([0,T])$ of the underlying Brownian motion centered at the constant function $m$ and radius $r$. The solution involves the derivation and solution of a related heat equation in which $f$ serves as a reaction term and which is accompanied by singular and discontinuous Dirichlet boundary conditions. Although the solution of the heat equation is discontinuous at the corners of the domain the BSDE has continuous sample paths with the prescribed terminal value.
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.
We prove, by a shooting method, the existence of infinitely many solutions of the form $psi(x^0,x) = e^{-iOmega x^0}chi(x)$ of the nonlinear Dirac equation {equation*} iunderset{mu=0}{overset{3}{sum}} gamma^mu partial_mu psi- mpsi - F(bar{psi}psi)psi = 0 {equation*} where $Omega>m>0,$ $chi$ is compactly supported and [F(x) = {{array}{ll} p|x|^{p-1} & text{if} |x|>0 0 & text{if} x=0 {array}.] with $pin(0,1),$ under some restrictions on the parameters $p$ and $Omega.$ We study also the behavior of the solutions as $p$ tends to zero to establish the link between these equations and the M.I.T. bag model ones.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
