We prove that $\frac{ \log n}{ n}$ is the sharp threshold for universality of the distribution of cokernels of random matrices over $\mathbb{ Z}_p$. More precisely, let $\alpha_n = \frac{ c\log n}{ n}$ for a constant $c>0$ and let $A(n)$ be an $\alpha_n$-balanced random matrix over $\mathbb{ Z}_p$. For non-symmetric, symmetric, and alternating matrix models, we prove that if $c>1$, then the limiting distribution of the cokernel of $A(n)$ coincides with the universal distribution of the corresponding symmetry type, whereas universality fails at the critical scale $c=1$. This improves earlier universality results, which required $\alpha_n \gg \frac{ \log n}{ n}$, to the optimal threshold. As an application, we generalize the universality result for Sylow $p$-subgroups of sandpile groups of Erd\H{ o}s-R\'enyi random graphs to a broader class of Erd\H{ o}s-R\'enyi graph sequences. Our approach is based on a unified framework that simultaneously treats all symmetry types of random matrices as well as the random graph model, rather than handling each case separately.
Categories: math.CO math.NT math.PR