  The walk matrix associated to an $n\times n$ integer matrix $X$ and an
integer vector $b$ is defined by $W := (b,X b, . . . ,X^{n-1} b)$. We study
limiting laws for the cokernel of $W$ in the scenario where $X$ is a random
matrix with independent entries and $b$ is deterministic. Our first main result
provides a formula for the distribution of the $p^{m}$-torsion part of the
cokernel, as a group, when $X$ has independent entries from a specific
distribution. The second main result relaxes the distributional assumption and
concerns the $\mathbb{Z}[x]$-module structure.
  The motivation for this work arises from an open problem in spectral graph
theory which asks to show that random graphs are often determined up to
isomorphism by their (generalized) spectrum. Sufficient conditions for
generalized spectral determinacy can namely be stated in terms of the cokernel
of a walk matrix. Extensions of our results could potentially be used to
determine how often those conditions are satisfied. Some remaining challenges
for such extensions are outlined in the paper
