Gaussian fluctuations of products of random matrices distributed close to the identity

Abstract

Products of random 2 £ 2 matrices exhibit Gaussian fluctuations around almost surely convergent Lyapunov exponents. In this paper, the distribution of the random matrices is supported by a small neighbourhood of order l . 0 of the identity matrix. The Lyapunov exponent and the variance of the Gaussian fluctuations are calculated perturbatively in l and this requires a detailed analysis of the associated random dynamical system on the unit circle and its invariant measure. The result applies to anomalies and band edges of one-dimensional random Schrödinger operators.

CLT with explicit variance for products of random singular matrices related to Hill's equation

“…Compared to λ, much less is known about σ 2 . A perturbative approach was used in [DSB15] to compute the leading order term in an asymptotic expansion for the variance of the product of random matrices which are close to the identity. In [MMP + 20], Monte Carlo simulation was used to approximate the variance in a specific 2 × 2 matrix model where λ is known exactly.…”

Representation theory and products of random matrices in $\text{SL}(2,{\mathbb R})$

“…This is an important mathematical object, relevant in several physical contexts: chaotic dynamics and multifractal analysis [8]; fluid dynamics [40]; classical dynamics of an oscillator driven by parametric noise [36,45]; polymers in a random landscape [17]. In particular, the generalised Lyapunov exponent yields information on the fluctuations of products of random matrices; these fluctuations are of interest in relation with Anderson localisation [10,11,34,37] and also in relation with disordered Ising spin chains [5].…”

Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents

“…This technique can be pushed to deal with large deviations [10] and a perturbative analysis of the variance in the central limit theorem [17]. When the rotation angle is trivial so that the real random 2 × 2 matrices are given by a random perturbation of the identity matrix, one deals with a so-called anomaly [11] and then the random phase property does not hold and alternative methods using second order ordinary differential equations on the unit circle have been developed to deal with this case [3,19,9]. Small random perturbations of a Jordan block are of relevance for band edges of random Jacobi matrices and can be dealt with by analyzing a singular differential equation [5,15].…”