We study a class of commuting Markov kernels whose simplest element describes the movement of $k$ particles on a discrete circle of size $n$ conditioned to not intersect each other. Such Markov kernels are related to the quantum cohomology ring of the Grassmannian, which is an algebraic object counting analytic maps from $\mathbb{P}^1(\mathbb{C})$ to the Grassmannian space of k-dimensional vector subspaces of $\mathbb{C}^n$ with prescribed constraints at some points of $\mathbb{P}^1(\mathbb{C})$. We obtain a Berry-Esseen theorem and a local limit theorem for an arbitrary product of approximately $n^2$ Markov kernels belonging to the above class, when k is fixed. As a byproduct of those results, we derive asymptotic formulas for the quantum cohomology ring of the Grassmannian in terms of the heat kernel on $SU (k)$.