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Random integral matrices: universality of surjectivity and the cokernel

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For a random matrix of entries sampled independently from a fairly general distribution in \({{\mathbf {Z}}}\) we study the probability that the cokernel is isomorphic to a given finite abelian group, or when it is cyclic. This includes the probability that the linear map between the integer lattices given by the matrix is surjective. We show that these statistics are asymptotically universal (as the size of the matrix goes to infinity), given by precise formulas involving zeta values, and agree with distributions defined by Cohen and Lenstra, even when the distribution of matrix entries is very distorted. Our method is robust and works for Laplacians of random digraphs and sparse matrices with the probability of an entry non-zero only \(n^{-1+\varepsilon }\).

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We would like to thank Wesley Pegden and Philip Matchett Wood for helpful conversations. We are also grateful to Nathan Kaplan, Lionel Levine and Philip Matchett Wood for valuable comments on an earlier version of this manuscript. Finally, we thank the anonymous referees for their careful reading of the paper and highly valuable suggestions and corrections. The first author is partially supported by National Science Foundation grants DMS-1600782 and NSF CAREER DMS-1752345. The second author is partially supported by a Packard Fellowship for Science and Engineering, a Sloan Research Fellowship, National Science Foundation grants DMS-1301690 and DMS-1652116, an NSF Waterman award, and a Vilas Early Career Investigator Award.

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Nguyen, H.H., Wood, M.M. Random integral matrices: universality of surjectivity and the cokernel. Invent. math. 228, 1–76 (2022).

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