TY - JOUR
T1 - Adjacency matrices of random digraphs
T2 - Singularity and anti-concentration
AU - Litvak, Alexander E.
AU - Lytova, Anna
AU - Tikhomirov, Konstantin
AU - Tomczak-Jaegermann, Nicole
AU - Youssef, Pierre
N1 - Publisher Copyright:
© 2016 Elsevier Inc.
PY - 2017/1/15
Y1 - 2017/1/15
N2 - Let Dn,d be the set of all d-regular directed 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 d-regular directed 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|≈n/d. Let δi be the indicator of the event that the vertex i is connected to J and define δ=(δ1,δ2,…,δn)∈{0,1}n. Then for every v∈{0,1}n the probability that δ=v is exponentially small. This property holds even if a part of the graph is “frozen.”
AB - Let Dn,d be the set of all d-regular directed 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 d-regular directed 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|≈n/d. Let δi be the indicator of the event that the vertex i is connected to J and define δ=(δ1,δ2,…,δn)∈{0,1}n. Then for every v∈{0,1}n the probability that δ=v is exponentially small. This property holds even if a part of the graph is “frozen.”
KW - Adjacency matrices
KW - Anti-concentration
KW - Invertibility of random matrices
KW - Littlewood–Offord theory
KW - Random regular graphs
KW - Singular probability
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U2 - 10.1016/j.jmaa.2016.08.020
DO - 10.1016/j.jmaa.2016.08.020
M3 - Article
AN - SCOPUS:84995393659
SN - 0022-247X
VL - 445
SP - 1447
EP - 1491
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 2
ER -