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Random 2D linear cocycles II: statistical properties

Pedro Duarte, Marcelo Durães, Tomé Graxinha, Silvius Klein

TL;DR

The paper addresses the behavior of random 2D linear cocycles with mixed singular and invertible components, focusing on Lyapunov exponents and their fluctuations. It delivers an explicit stationary measure for the associated projective Markov chain and a Furstenberg-type formula for the first Lyapunov exponent $L_1$, derived via a desingularized Markov operator and a detailed partition of the state space. Using spectral properties and a parameter-elimination strategy, it proves sub-exponential large deviations and a central limit theorem for Lebesgue-a.e. cocycles in the admissible family $ extcal{M}$, including parametric versions along one-parameter families with positive winding. The results extend known invertible-case theory to the singular+invertible setting, with implications for random Schrödinger-type operators and a near-complete understanding of $L_1$ regularity and fluctuation laws in finite-alphabet, 2D cocycles.

Abstract

Consider the space of two dimensional random linear cocycles over a shift in finitely many symbols, with at least one singular and one invertible matrix. We provide an explicit formula for the unique stationary measure associated to such cocycles and establish a Furstenberg-type formula characterizing the Lyapunov exponent. Using the spectral properties of the corresponding Markov operator and a parameter elimination argument, we prove that Lebesgue almost every cocycle in this space satisfies large deviations estimates and a central limit theorem.

Random 2D linear cocycles II: statistical properties

TL;DR

The paper addresses the behavior of random 2D linear cocycles with mixed singular and invertible components, focusing on Lyapunov exponents and their fluctuations. It delivers an explicit stationary measure for the associated projective Markov chain and a Furstenberg-type formula for the first Lyapunov exponent , derived via a desingularized Markov operator and a detailed partition of the state space. Using spectral properties and a parameter-elimination strategy, it proves sub-exponential large deviations and a central limit theorem for Lebesgue-a.e. cocycles in the admissible family , including parametric versions along one-parameter families with positive winding. The results extend known invertible-case theory to the singular+invertible setting, with implications for random Schrödinger-type operators and a near-complete understanding of regularity and fluctuation laws in finite-alphabet, 2D cocycles.

Abstract

Consider the space of two dimensional random linear cocycles over a shift in finitely many symbols, with at least one singular and one invertible matrix. We provide an explicit formula for the unique stationary measure associated to such cocycles and establish a Furstenberg-type formula characterizing the Lyapunov exponent. Using the spectral properties of the corresponding Markov operator and a parameter elimination argument, we prove that Lebesgue almost every cocycle in this space satisfies large deviations estimates and a central limit theorem.
Paper Structure (7 sections, 23 theorems, 160 equations, 1 table)

This paper contains 7 sections, 23 theorems, 160 equations, 1 table.

Table of Contents

  1. Introduction and statements
  2. Stationary measures
  3. Statistical Properties
  4. Assumptions
  5. Some technical estimates
  6. Large deviations estimates
  7. Central limit theorem

Key Result

Theorem 1.1

For Lebesgue almost every random cocycle $\underline{A}\in \mathcal{M}$, $L_1(\underline{A})> - \infty$ and for every $\varepsilon>0$ we have where $C< \infty$, $c_0 (\varepsilon)>0$ is an explicit function of $\varepsilon$ and $\mu$ is the Bernoulli (or Markov) measure on $X$ defined above.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.1
  • Proposition 2.1
  • proof
  • Theorem 2.1
  • proof
  • proof
  • Corollary 2.1
  • ...and 45 more