[Paper Review] Random products of matrices: a dynamical point of view

Tien‐Cuong Dinh, Lucas Kaufmann+1 more|arXiv (Cornell University)|May 21, 2019

Geometry and complex manifolds9 references4 citations

TL;DR

This paper studies random matrix products in SL₂(ℂ) using holomorphic dynamics, establishing exponential convergence of the transfer operator to the stationary measure in Sobolev norm under finite first moment conditions. It proves exponentially fast equidistribution of forward orbits and provides a new proof of the Central Limit Theorem for the norm cocycle under second moment conditions, with applications to regularity of stationary measures.

ABSTRACT

We study random products of matrices in SL_2(C) from the point of view of holomorphic dynamics. For non-elementary measures with finite first moment we obtain the exponential convergence towards the stationary measure in Sobolev norm. As a consequence we obtain the exponentially fast equidistribution of forward images of points towards the stationary measure. We also give a new proof of the Central Limit Theorem for the norm cocycle under a second moment condition, originally due to Benoist-Quint, and obtain some general regularity results for stationary measures.

Motivation & Objective

• To analyze random products of SL₂(ℂ) matrices using tools from holomorphic dynamics.
• To establish exponential convergence of the transfer operator to the stationary measure in Sobolev norm under finite first moment conditions.
• To prove exponentially fast equidistribution of forward images of points toward the stationary measure.
• To give a new proof of the Central Limit Theorem for the norm cocycle under second moment conditions.
• To establish regularity properties of stationary measures under moment conditions.

Proposed method

• Use of the generalized correspondence $ f_\mu $ on $ \mathbb{P}^1 \times \mathbb{P}^1 $ defined by the current $ [\Gamma_\mu] $, integrating over the group action.
• Application of the transfer operator $ f^*_\mu $ defined by $ f^*_\mu(\varphi)(x) = \int_G \varphi(g \cdot x) \, d\mu(g) $, acting on test functions in $ \mathcal{C}^\beta $ and Sobolev spaces.
• Employment of Sobolev norm $ \|\cdot\|_{W^{1,2}} $ to quantify convergence of the transfer operator to the stationary measure $ \nu $.
• Use of Cartan decomposition to reduce analysis to diagonal matrices, enabling explicit computation of $ \|\partial\theta_g\|_{L^2} $ and $ \|\theta_g\|_{W^{1,2}} $.
• Proof of exponential decay via a lemma showing that if $ g^n h^n $ is not loxodromic for infinitely many $ n $, then $ g $ and $ h $ share a fixed point, contradicting genericity.
• Use of Lemma A.7 to bound the $ L^2 $-norm of the differential of $ \theta_g $, leading to $ \|\theta_g\|_{W^{1,2}} \lesssim 1 + \log\|g\| $.

Experimental results

Research questions

• RQ1 Under what moment conditions does the transfer operator converge exponentially fast to the stationary measure in Sobolev norm?
• RQ2 How fast do forward images of points equidistribute toward the stationary measure under non-elementary measures?
• RQ3 Can the Central Limit Theorem for the norm cocycle be re-proven using holomorphic dynamics and Sobolev estimates?
• RQ4 What regularity properties does the stationary measure possess under finite first or second moment conditions?
• RQ5 What conditions ensure that iterated products $ g_n \cdots g_1 $ are loxodromic for large $ n $, enabling ergodicity and mixing?

Key findings

• For non-elementary $ \mu \in \mathrm{PSL}_2(\mathbb{C}) $ with finite first moment, the transfer operator converges exponentially fast to the stationary measure $ \nu $ in $ W^{1,2} $-norm.
• Forward images of any point $ a \in \mathbb{P}^1 $ equidistribute toward $ \nu $ with exponential speed $ \gamma^{\beta n} $, uniformly in $ a $, under $ \int (\log\|g\|)^{1+\epsilon} \, d\mu(g) < \infty $.
• A new proof of the Central Limit Theorem for the norm cocycle is obtained under the second moment condition $ \int \|g\|^\alpha \, d\mu(g) < \infty $ for some $ \alpha > 0 $, originally due to Benoist-Quint.
• The stationary measure $ \nu $ is shown to be regular under moment conditions, with quantitative control on the growth of $ \|\theta_g\|_{W^{1,2}} \lesssim 1 + \log\|g\| $.
• The proof establishes that if $ g^n h^n $ is not loxodromic for infinitely many $ n $, then $ g $ and $ h $ must share a fixed point, which is ruled out under non-elementarity.
• The $ L^2 $-norm of the differential of $ \theta_g $ is bounded by $ \lesssim (\log\|g\|)^{1/2} $, enabling the Sobolev estimate and subsequent convergence results.