Universality of the cokernels of random $p$-adic matrices with inhomogeneously balanced columns

In this paper, we prove universality of the distribution of the cokernels of a random $p$-adic matrix with inhomogeneously balanced columns. More precisely, let $u\ge 0$ be an integer and $A(n)$ be a random $n\times (n+u)$ matrix over ${\mathbb{Z}}_p$ whose $i$-th column is ${\alpha }_n(i)$-balanced. We prove that if ${\sum }_{i=1}^{n+u}\exp (-\epsilon {\alpha }_n(i)n)\to 0$ as $n\to \mathrm{\infty }$ for every $\epsilon >0$, then the cokernels of $A(n)$ converge in distribution, as $n\to \mathrm{\infty }$, to the same limiting law as the cokernels of Haar-random $n\times (n+u)$ matrices over ${\mathbb{Z}}_p$. This extends a universality theorem of Nguyen and Wood to random $p$-adic matrices with inhomogeneously balanced columns.