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The sparse circular law, revisited
- Publication Year :
- 2023
-
Abstract
- Let $A_n$ be an $n\times n$ matrix with iid entries distributed as Bernoulli random variables with parameter $p = p_n$. Rudelson and Tikhomirov, in a beautiful and celebrated paper, show that the distribution of eigenvalues of $A_n \cdot (pn)^{-1/2}$ is approximately uniform on the unit disk as $n\rightarrow \infty$ as long as $pn \rightarrow \infty$, which is the natural necessary condition. In this paper we give a much simpler proof of this result, in its full generality, using a perspective we developed in our recent proof of the existence of the limiting spectral law when $pn$ is bounded. One feature of our proof is that it avoids the use of $\epsilon$-nets entirely and, instead, proceeds by studying the evolution of the singular values of the shifted matrices $A_n-zI$ as we incrementally expose the randomness in the matrix.<br />Comment: 20 pages
- Subjects :
- Mathematics - Probability
Mathematics - Combinatorics
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2310.17600
- Document Type :
- Working Paper