@article{02e240ff7fcf4c38b9fb822dcdb7907f,

title = "Circular law for sparse random regular digraphs",

abstract = "Fix a constant C ≥ 1 and let d = d(n) satisfy d ≤ lnC n for every large integer n. Denote by An the adjacency matrix of a uniform random directed d-regular graph on n vertices. We show that if d → ∞ as n → ∞, the empirical spectral distribution of the appropriately rescaled matrix An converges weakly in probability to the circular law. This result, together with an earlier work of Cook, completely settles the problem of weak convergence of the empirical distribution in a directed d-regular setting with the degree tending to infinity. As a crucial element of our proof, we develop a technique of bounding intermediate singular values of An based on studying random normals to rowspaces and on constructing a product structure to deal with the lack of independence between matrix entries. ",

keywords = "Circular law, Intermediate singular values, Logarithmic potential, Random graphs, Random matrices, Regular graphs, Sparse matrices",

author = "Litvak, {Alexander E.} and Anna Lytova and Konstantin Tikhomirov and Nicole Tomczak-Jaegermann and Pierre Youssef",

note = "Funding Information: Acknowledgments. P.Y. was supported by grant ANR-16-CE40-0024-01. A.L. was supported by grant no. 2018/31/B/ST1/03937, National Science Centre, Poland. A significant part of this work was completed while the last three named authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, supported by NSF grant DMS-1440140, and the first two named authors visited the Institute. The hospitality of MSRI and of the organizers of the program on Geometric Functional Analysis and Applications is gratefully acknowledged. Funding Information: P.Y. was supported by grant ANR-16-CE40-0024-01. A.L. was supported by grant no. 2018/31/B/ST1/03937, National Science Centre, Poland. A significant part of this work was completed while the last three named authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, supported by NSF grant DMS-1440140, and the first two named authors visited the Institute. The hospitality of MSRI and of the organizers of the program on Geometric Functional Analysis and Applications is gratefully acknowledged. Publisher Copyright: {\textcopyright} European Mathematical Society 2021.",

year = "2021",

month = oct,

day = "29",

doi = "10.4171/JEMS/1015",

language = "English (US)",

volume = "23",

pages = "467--501",

journal = "Journal of the European Mathematical Society",

issn = "1435-9855",

publisher = "European Mathematical Society Publishing House",

number = "2",

}