TY - JOUR

T1 - Anti-concentration property for random digraphs and invertibility of their adjacency matrices

AU - Litvak, Alexander E.

AU - Lytova, Anna

AU - Tikhomirov, Konstantin

AU - Tomczak-Jaegermann, Nicole

AU - Youssef, Pierre

N1 - Funding Information:
This work was conducted while the last named author was a CNRS/PIMS PDF at the University of Alberta. All authors would like to thank the university for the excellent working conditions.
Publisher Copyright:
© 2015 Académie des sciences.

PY - 2016/2/1

Y1 - 2016/2/1

N2 - 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.

AB - 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.

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U2 - 10.1016/j.crma.2015.12.002

DO - 10.1016/j.crma.2015.12.002

M3 - Article

AN - SCOPUS:84956845390

VL - 354

SP - 121

EP - 124

JO - Comptes Rendus Mathematique

JF - Comptes Rendus Mathematique

SN - 1631-073X

IS - 2

ER -