Anti-concentration property for random digraphs and invertibility of their adjacency matrices

Alexander E. Litvak, Anna Lytova, Konstantin Tikhomirov, Nicole Tomczak-Jaegermann, Pierre Youssef

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Abstract

Let Dn,d be the set of all directed d-regular graphs on n vertices. Let G be a graph chosen uniformly at random from Dn,d and M be its adjacency matrix. We show that M is invertible with probability at least 1-Cln3d/d for C≤d≤cn/ln2n, where c, C are positive absolute constants. To this end, we establish a few properties of directed d-regular graphs. One of them, a Littlewood-Offord-type anti-concentration property, is of independent interest: let J be a subset of vertices of G with |J|≤cn/d. Let δi be the indicator of the event that the vertex i is connected to J and δ=(δ1, δ2, . ., δn)∈[0, 1]n. Then δ is not concentrated around any vertex of the cube. This property holds even if a part of the graph is fixed.

Original languageEnglish (US)
Pages (from-to)121-124
Number of pages4
JournalComptes Rendus Mathematique
Volume354
Issue number2
DOIs
StatePublished - Feb 1 2016

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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    Litvak, A. E., Lytova, A., Tikhomirov, K., Tomczak-Jaegermann, N., & Youssef, P. (2016). Anti-concentration property for random digraphs and invertibility of their adjacency matrices. Comptes Rendus Mathematique, 354(2), 121-124. https://doi.org/10.1016/j.crma.2015.12.002