• Corpus ID: 218570964

A Local Law for Singular Values from Diophantine Equations

@article{Adhikari2020ALL,
  title={A Local Law for Singular Values from Diophantine Equations},
  author={Arka Adhikari and Marius Lemm},
  journal={arXiv: Probability},
  year={2020}
}
We introduce the $N\times N$ random matrices $$ X_{j,k}=\exp\left(2\pi i \sum_{q=1}^d\ \omega_{j,q} k^q\right) \quad \text{with } \{\omega_{j,q}\}_{\substack{1\leq j\leq N\\ 1\leq q\leq d}} \text{ i.i.d. random variables}, $$ 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 $d\geq 18$. To our knowledge, this is the first instance of a random… 

Universal Eigenvalue Statistics for Dynamically Defined Matrices

We consider dynamically defined Hermitian matrices generated from orbits of the doubling map. We prove that their spectra fall into the GUE universality class from random matrix theory.

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