Convergence of permuted products of exponentials
Michael Anshelevich, Anh Nguyen
TL;DR
This work investigates the convergence of permuted products of exponentials ∏_{i=1}^{[tn]} e^{A_{σ(i),n}/n} where {A_{i,n}} is a triangular array with row averages converging to A in a Banach algebra. The authors show that, under mild combinatorial constraints (o(n/log n) distinct elements per row) or in the finite-dimensional matrix case (where the restriction is unnecessary), random rearrangements yield uniform convergence of the path to e^{tA} for t∈[0,1], with convergence guaranteed almost surely or in probability depending on the setting. The analysis combines translating multiplicative convergence to additive convergence and applying concentration inequalities (in the matrix setting) to obtain quantitative bounds, including a Lie–Trotter-type result. In the matrix case, explicit growth conditions on L^1_n and L^∞_n are provided (e.g., (L^1_n)^{4+δ} e^{3 L^1_n} L^ abla_n = o(n) for convergence in probability and (L^1_n)^3 e^{3 L^1_n} L^ abla_n = o(n/ log n) for almost sure convergence), along with discussions of time-ordered exponentials and randomized evolution families. These results advance understanding of randomized evolutions and product integrals in both abstract Banach-algebra and finite-dimensional contexts.
Abstract
Let $\{A_{i,n}\}$ be a triangular array of elements in a Banach algebra, whose norms do not grow too fast, and whose row averages converge to $A$. Let $σ\in S(n)$ be a permutation drawn uniformly at random. If the array only contains $o(n / \log n)$ distinct elements, then almost surely, for all $0 < s < t < 1$, the permuted product of their exponentials $\prod_{i = [s n]}^{[t n]} e^{A_{σ(i),n}/n}$ converges in norm to $e^{(t - s) A}$. For an array of finite-dimensional matrices, convergence holds without this restriction. The proof of the latter result consists of an estimate valid in a general Banach algebra, and an application of a matrix concentration inequality.