Abstract
We prove a lower bound expansion on the probability that a random ±1 matrix is singular, and conjecture that such expansions govern the actual probability of singularity. These expansions are based on naming the most likely, second most likely, and so on, ways that a Bernoulli matrix can be singular; the most likely way is to have a null vector of the form e i ±e j , which corresponds to the integer partition 11, with two parts of size 1. The second most likely way is to have a null vector of the form e i ±e j ±e k ±e ℓ , which corresponds to the partition 1111. The fifth most likely way corresponds to the partition 21111. We define and characterize the “novel partitions” which show up in this series. As a family, novel partitions suffice to detect singularity, i.e., any singular Bernoulli matrix has a left null vector whose underlying integer partition is novel. And, with respect to this property, the family of novel partitions is minimal. We prove that the only novel partitions with six or fewer parts are 11, 1111, 21111, 111111, 221111, 311111, and 322111. We prove that there are fourteen novel partitions having seven parts. We formulate a conjecture about which partitions are “first place and runners up” in relation to the Erdős-Littlewood-Offord bound. We prove some bounds on the interaction between left and right null vectors.
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Alon, N., Kozlov, D.N., Vu, V.H.: The geometry of coin-weighing problems. In: 37th Annual Symposium on Foundations of Computer Science, pp. 524–532. IEEE Comput. Soc. Press, Los Alamitos, CA (1996)
Bollobás B.: Random Graphs. 2nd Edit. Cambridge University Press, Cambridge (2001)
Bourgain J., Vu V.H., Wood P.M.: On the singularity probability of discrete random matrices. J. Funct. Anal. 258(2), 559–603 (2010)
Erdös P.: On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc. 51, 898–902 (1945)
Kahn J., Komlós J., Szemerédi E.: On the probability that a random ±1-matrix is singular. J. Amer. Math. Soc. 8(1), 223–240 (1995)
Komlós, J.: Circulated manuscript. Unpublished (1977)
Mathematica. Mathematica Edition: Version 8.0. Wolfram Research, Inc., Champaign, IL (2010)
Metropolis N., Stein P.R.: On a class of (0, 1) matrices with vanishing determinants. J. Combin. Theory 3, 191–198 (1967)
Odlyzko A.M.: On subspaces spanned by random selections of ±1 vectors. J. Combin. Theory Ser. A 47(1), 124–133 (1988)
Tao T., Vu V.: On random ±1 matrices: singularity and determinant. Random Structures Algorithms 28(1), 1–23 (2006)
Tao T., Vu V.: On the singularity probability of random Bernoulli matrices. J. Amer. Math. Soc. 20(3), 603–628 (2007)
Tao T., Vu V.H.: Additive Combinatorics. Cambridge University Press, Cambridge (2010)
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Arratia, R., DeSalvo, S. On the Singularity of Random Bernoulli Matrices— Novel Integer Partitions and Lower Bound Expansions. Ann. Comb. 17, 251–274 (2013). https://doi.org/10.1007/s00026-013-0176-7
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DOI: https://doi.org/10.1007/s00026-013-0176-7