Limit theorems for a strongly irreducible product of independent random matrices under optimal moment assumptions
Axel Péneau
TL;DR
This work analyzes products of i.i.d. random matrices under proximal and strongly irreducible hypotheses, without relying on classical finite-moment assumptions. It introduces a Markovian extraction framework and a pivoting technique to group factors into aligned words in the Cartan projection, yielding precise control of the linear escape rate of the logarithmic singular gap and exponential large deviations below the escape rate. The paper proves that the image of a generic line and its maximal-eigenvalue eigenspace converge to a random line l_∞ with exponential speed, and, under a finite $p$-th moment for the push-forward distribution of the Cartan functional, that log-coefficients along l_∞ and associated linear forms are in $L^p$. It also develops a contraction/regularity theory via exterior powers, constructs Schottky measures, and derives a full suite of local-to-global alignment results, limit flags, and LLNs for coefficients and spectral radius, significantly extending random-product theory beyond traditional GL-server moment conditions and providing quantitative, algorithmic alignment tools.
Abstract
Let $ ν$ be a probability distribution over the linear semi-group $ \mathrm{End}(E) $ for $ E $ a finite dimensional vector space over a locally compact field. We assume that $ ν$ is proximal, strongly irreducible and that $ ν^{*n}\{0\}=0 $ for all integers $ n\in\mathbb{N} $. We consider the random sequence $ \overlineγ_n := γ_0 \cdots γ_{n-1} $ for $ (γ_k)_{k \ge 0} $ independents of distribution law $ ν$. We define the logarithmic singular gap as $ \mathrm{sqz} = \log\left( \frac{μ_1}{μ_2} \right) $ , where $ μ_1 $ and $ μ_2 $ are the two largest singular values. We show that $ (\mathrm{sqz}(\overlineγ_n))_{n\in\mathbb{N}} $ escapes to infinity linearly and satisfies exponential large deviations estimates below its escape rate. With the same assumptions, we also show that the image of a generic line by $ \overlineγ_n $ as well as its eigenspace of maximal eigenvalue both converge to the same random line $l_\infty $ at an exponential speed.If we moreover assume that the push-forward distribution $N(ν)$ is $ \mathrm{L}^p $ for $ N:g\mapsto\log\left(\|g\|\|g^{-1}\|\right) $ and for some $ p\ge 1 $, then we show that $ \log|w(l_\infty)| $ is $ \mathrm{L}^p $ for all unitary linear form $ w $ and the logarithm of each coefficient of $ \overlineγ_n $ is almost surely equivalent to the logarithm of the norm. To prove these results, we do not rely on any classical results for random products of invertible matrices with $ \mathrm{L}^1 $ moment assumption. Instead we describe an effective way to group the i.i.d factors into i.i.d random words that are aligned in the Cartan projection. We moreover have an explicit control over the moments.