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Approximate Spielman-Teng theorems for the least singular value of random combinatorial matrices

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Abstract

An approximate Spielman-Teng theorem for the least singular value sn(Mn) of a random n × n square matrix Mn is a statement of the following form: there exist constants C, c > 0 such that for all η > 0, Pr(sn(Mn) ≤ η) ≲ nCη + exp(−nc). The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for {0, 1}-valued matrices, each of whose rows is an independent vector with exactly n/2 zero components. This improves on previous work of Nguyen and Vu, and is the first such result in a ‘truly combinatorial’ setting.

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Acknowledgements

I would like to thank Kyle Luh for comments on a preliminary version of this paper, and Jake Lee Wellens for helpful conversations. I am also grateful to the anonymous referee for careful reading of the manuscript and numerous valuable comments.

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  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Vishesh Jain

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  1. Vishesh Jain
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Jain, V. Approximate Spielman-Teng theorems for the least singular value of random combinatorial matrices. Isr. J. Math. 242, 461–500 (2021). https://doi.org/10.1007/s11856-021-2144-y

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  • DOI: https://doi.org/10.1007/s11856-021-2144-y

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