• 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}
}
• Published 8 May 2020
• Mathematics
• arXiv: Probability
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…
1 Citations

### Universal Eigenvalue Statistics for Dynamically Defined Matrices

• Mathematics
• 2022
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.

## References

SHOWING 1-10 OF 33 REFERENCES

• Mathematics
• 2009