Product of exponentials concentrates around the exponential of the sum

2022/07/01 by Anshelevich, Michael, Pritchett, Austin
#Combinatorics (math.CO) #FOS: Mathematics #Primary 15A16 #Probability (math.PR) #Rings and Algebras (math.RA) #Secondary 05A16

paper · doi:10.48550/arxiv.2207.00613

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.

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