  In this paper we study the cokernels of various random integral matrix
models, including random symmetric, random skew-symmetric, and random Laplacian
matrices. We provide a systematic method to establish universality under very
general randomness assumption. Our highlights include both local and global
universality of the cokernel statistics of all these models. In particular, we
find the probability that a sandpile group of an Erdos-Renyi random graph is
cyclic, answering a question of Lorenzini from 2008.
