Circular law for sparse random regular digraphs
Alexander E. Litvak
University of Alberta, Edmonton, CanadaAnna Lytova
University of Opole, PolandKonstantin Tikhomirov
Princeton University, USANicole Tomczak-Jaegermann
University of Alberta, Edmonton, CanadaPierre Youssef
Université Paris-Diderot, France
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Abstract
Fix a constant and let satisfy for every large integer . Denote by the adjacency matrix of a uniform random directed -regular graph on vertices. We show that if as , the empirical spectral distribution of the appropriately rescaled matrix 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 -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 based on studying random normals to rowspaces and on constructing a product structure to deal with the lack of independence between matrix entries.
Cite this article
Alexander E. Litvak, Anna Lytova, Konstantin Tikhomirov, Nicole Tomczak-Jaegermann, Pierre Youssef, Circular law for sparse random regular digraphs. J. Eur. Math. Soc. 23 (2021), no. 2, pp. 467–501
DOI 10.4171/JEMS/1015