Random integral matrices: universality of surjectivity and the cokernel

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