Products of random matrices: Dimension and growth in norm

TL;DR

This paper investigates the convergence rate of the normalized logarithm of the norm of products of i.i.d. rotationally invariant matrices as the matrix size increases, establishing conditions under which the convergence speed remains unaffected by dimension.

Contribution

It proves that under specific assumptions, the convergence speed of the norm growth rate does not diminish as the matrix dimension increases.

Findings

Convergence speed remains stable with increasing matrix size.

Established conditions for dimension-independent convergence rate.

Enhanced understanding of growth behavior in high-dimensional random matrix products.

Abstract

Suppose that $X_1,\...,X_n,\...$ are i.i.d. rotationally invariant $N$-by-$N$ matrices. Let $\Pi_n=X_n\... X_1$. It is known that $n^{-1}\log |\Pi_n|$ converges to a nonrandom limit. We prove that under certain additional assumptions on matrices $X_i$ the speed of convergence to this limit does not decrease when the size of matrices, $N$, grows.