The distribution of the cokernel of a polynomial evaluated at a random integral matrix

Given a prime $p$, let $P(t)$ be a non-constant monic polynomial in $t$ over the ring ${\mathbb{Z}}_p$ of $p$-adic integers. Let $X_n$ be an $n\times n$ random matrix over ${\mathbb{Z}}_p$ with independent entries that lie in any residue class modulo $p$ with probability at most $1-\epsilon$ for a fixed real number $0<\epsilon <1$. We prove that as $n\to \mathrm{\infty }$, the distribution of the cokernel $\mathrm{c}\mathrm{o}\mathrm{k}(P(X_n))$ of $P(X_n)$ converges to the distribution given by a finite product of some explicit measures that resemble Cohen–Lenstra measures. For example, the random matrix $X_n$ can be taken as a Haar-random matrix or a uniformly random $(0,1)$-matrix. We consider the distribution of $\mathrm{c}\mathrm{o}\mathrm{k}(P(X_n))$ as a distribution of modules over ${\mathbb{Z}}_p[t]\mathrm{/}(P(t))$, which gives us a clearer formulation in comparison to considering the distribution as that of abelian groups. For the proof, we first reduce our problem into a problem over $\mathbb{Z}\mathrm{/}{p}^k\mathbb{Z}$, for large enough positive integer $k$, in place of ${\mathbb{Z}}_p$. Then we use a result of Sawin and Wood to reduce our problem into another problem of computing the limit of the expected number of surjective $(\mathbb{Z}\mathrm{/}{p}^k\mathbb{Z})[t]\mathrm{/}(P(t))$-linear maps from $\mathrm{c}\mathrm{o}\mathrm{k}(P(X_n))$ modulo $p^k$ to a fixed finite size $(\mathbb{Z}\mathrm{/}{p}^k\mathbb{Z})[t]\mathrm{/}(P(t))$-module $G$. To estimate the expected number and compute the desired limit, we carefully adopt subtle techniques developed by Wood, which were originally used to compute the asymptotic distribution of the $p$-part of the sandpile group of a random graph.