Abstract
We prove the first eigenvalue repulsion bound for sparse random matrices. As a consequence, we show that these matrices have simple spectrum, improving the range of sparsity and error probability from work of the second author and Vu. We also show that for sparse Erdős–Rényi graphs, weak and strong nodal domains are the same, answering a question of Dekel, Lee, and Linial.
Funding Statement
K. Luh was partially supported by NSF postdoctoral fellowship DMS-1702533. P.L. was partially supported by the NSF Graduate Research Fellowship Program under grant DGE-1144152.
Acknowledgments
The authors thank the anonymous referees for their detailed comments, which substantially improved the paper.
Citation
Patrick Lopatto. Kyle Luh. "Tail bounds for gaps between eigenvalues of sparse random matrices." Electron. J. Probab. 26 1 - 26, 2021. https://doi.org/10.1214/21-EJP669
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