Product of exponentials concentrates around the exponential of the sum

TL;DR

This paper demonstrates that for large products of exponentials of matrices, the product concentrates around the exponential of the sum, extending the Lie-Trotter formula with a new elementary proof.

Contribution

It provides a new elementary proof showing concentration of matrix exponential products around the exponential of the sum, extending to multiple matrices.

Findings

Most products of $e^{A/n}$ and $e^{B/n}$ are close to $e^{A+B}$ for large n

The proof uses lattice paths, binomial asymptotics, and matrix inequalities

The result generalizes to more than two matrices

Abstract

For two matrices $A$ and $B$, and large $n$, we show that most products of $n$ factors of $e^{A/n}$ and $n$ factors of $e^{B/n}$ are close to $e^{A + B}$. This extends the Lie-Trotter formula. The elementary proof is based on the relation between words and lattice paths, asymptotics of binomial coefficients, and matrix inequalities. The result holds for more than two matrices.