The smallest singular value of a shifted d-regular random square matrix

@article{Litvak2018TheSS,
title={The smallest singular value of a shifted d-regular random square matrix},
author={Alexander E. Litvak and Anna Lytova and Konstantin E. Tikhomirov and Nicole Tomczak-Jaegermann and Pierre Youssef},
journal={Probability Theory and Related Fields},
year={2018},
volume={173},
pages={1301-1347}
}
We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let $$C_1<d< c n/\log ^2 n$$C1<d<cn/log2n and let $$\mathcal {M}_{n,d}$$Mn,d be the set of all $$n\times n$$n×n square matrices with 0 / 1 entries, such that each row and each column of every matrix in $$\mathcal {M}_{n,d}$$Mn,d has exactly d ones. Let M be a random matrix uniformly distributed on $$\mathcal {M}_{n,d}$$Mn,d. Then…
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