Arithmetic Properties of Random Matrices

The singularity problem for random matrices over the integers is generalized to compute the rank of a random matrix modulo a prime, and it is shown that the rank distribution converges to the expected distribution at an exponential rate. Similarly, when the matrix is embedded into the p-adic integers, it is shown that the cokernel is distributed according to the Cohen-Lenstra heuristics. This limit holds for any iid random matrix subject to minor distributional requirements. Similar results are found for matrices over composite bases.;A new type of Littlewood-Offord theorem is developed for random sums in torsion groups. This theorem shows that such sums that do not distribute uniformly over the group must be structured. This is used to study the probability that a random vector lies in a random submodule.;These results imply an analogue of the Brun-Titchmarsh theorem for the determinant of a random matrix over the integers.