论文标题
指数的产物集中在总和的指数周围
Product of exponentials concentrates around the exponential of the sum
论文作者
论文摘要
对于两个矩阵$ a $和$ b $，以及大$ n $，我们表明$ e^{a/n} $的大多数产品和$ e^{b/n} $的$ e^{a/n $ factor of $ e^{b/n} $都接近$ e^{a + b} $。这扩展了Lie-Trotter公式。基本证明是基于单词和晶格路径之间的关系，二项式系数的渐近学以及基质不等式的。结果可容纳两个以上的矩阵。
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.
