Surjectivity of near-square random matrices

We show that a nearly square iid random integral matrix is surjective over the integral lattice with very high probability. This answers a question by Koplewitz. Our result extends to sparse matrices as well as to matrices of dependent entries.

[1]  Melanie Matchett Wood,et al.  The distribution of sandpile groups of random graphs , 2014, 1402.5149.

[2]  Melanie Matchett Wood,et al.  Random integral matrices and the Cohen-Lenstra heuristics , 2015, American Journal of Mathematics.

[3]  Jean Bourgain,et al.  On the singularity probability of discrete random matrices , 2009, 0905.0461.

[4]  Roman Vershynin,et al.  Invertibility of symmetric random matrices , 2011, Random Struct. Algorithms.

[5]  K. Maples.  Arithmetic properties of random matrices , 2012 .

[6]  Mark Rudelson,et al.  Invertibility of Sparse non-Hermitian matrices , 2015, 1507.03525.

[7]  G. Halász.  Estimates for the concentration function of combinatorial number theory and probability , 1977 .

[8]  Terence Tao,et al.  Additive combinatorics , 2007, Cambridge studies in advanced mathematics.

[9]  Hoi H. Nguyen,et al.  Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices , 2011, 1101.3074.

[10]  E. Szemerédi,et al.  On the probability that a random ±1-matrix is singular , 1995 .

[11]  Kenneth Maples,et al.  Cokernels of random matrices satisfy the Cohen-Lenstra heuristics , 2013, 1301.1239.

[12]  R. Stanley.  What Is Enumerative Combinatorics , 1986 .

[13]  Hoi H. Nguyen,et al.  Random matrices: Overcrowding estimates for the spectrum , 2017, Journal of Functional Analysis.

[14]  M. Rudelson,et al.  The Littlewood-Offord problem and invertibility of random matrices , 2007, math/0703503.

[15]  T. Tao,et al.  On the singularity probability of random Bernoulli matrices , 2005, math/0501313.

[16]  The corank of a rectangular random integer matrix , 2016, Linear Algebra and its Applications.

[17]  K. Maples.  Singularity of Random Matrices over Finite Fields , 2010, 1012.2372.