The sparse circular law, revisited

2023/10/26 by Ashwin Sah, Julian Sahasrabudhe, Sah, Ashwin +3 · 3 citations
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2310.17600

openalex publication_date 2023/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A_n be an n× 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 ⋅ (pn)^-1/2 is approximately uniform on the unit disk as n→ ∞ as long as pn → ∞, 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 ε-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.

Cited by

• Probabilistic combinatorics at exponentially small scales
• The Limiting Spectral Distribution for Sparse Elliptic Random Matrices
• Invertibility for non-Hermitian and symmetric random band matrices with sublinear bandwidth and discrete entries

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