Quantum cohomology of the Grassmannian and unitary Dyson Brownian motion

TL;DR

This paper explores the connection between quantum cohomology of the Grassmannian and unitary Dyson Brownian motion, providing limit theorems and asymptotic formulas for related Markov kernels and algebraic structures.

Contribution

It introduces a class of commuting Markov kernels linked to the quantum cohomology ring of the Grassmannian and derives limit theorems and asymptotic formulas for these kernels.

Findings

Berry-Esseen theorem established for the kernels

Local limit theorem proved for product of kernels

Asymptotic formulas for quantum cohomology in terms of heat kernel on SU(k)

Abstract

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 $SU (k)$.