Related papers: Random and mean Lyapunov exponents for $\mathrm{GL…
Analyticity and other properties of the largest or smallest Lyapunov exponent of a product of real matrices with a "cone property" are studied as functions of the matrices entries, as long as they vary without destroying the cone property.…
We study the asymptotic behavior of the Lyapunov exponent in a meromorphic family of random products of matrices in SL(2, C), as the parameter converges to a pole. We show that the blow-up of the Lyapunov exponent is governed by a quantity…
This article presents maximum likelihood estimators (MLEs) and log-likelihood ratio (LLR) tests for the eigenvalues and eigenvectors of Gaussian random symmetric matrices of arbitrary dimension, where the observations are independent…
We derive a series summation formula for the average logarithm norm of the action of a matrix on the projective space. This formula is shown to be useful to evaluate some Lyapunov exponents of random $\SL$-matrix cocycles, which include a…
The Lyapunov exponent corresponding to a set of square matrices $\mathcal{A} = \{A_1, \dots, A_n \}$ and a probability distribution $p$ over $\{1, \dots, n\}$ is $\lambda(\mathcal{A},p) := \lim_{k \to \infty} \frac{1}{k} \,\mathbb{E} \log…
We show that the convolution of a compactly supported measure on $\mathbb{R}$ with a Gaussian measure satisfies a logarithmic Sobolev inequality (LSI). We use this result to give a new proof of a classical result in random matrix theory…
The Lyapunov exponents of GL(2)-cocycles over Markov shifts depend continuously on the underlying data, that is, on the matrix coefficients and the Markov measure transition probabilities.
Ergodic properties of rational maps are studied, generalising the work of F.\ Ledrappier. A new construction allows for simpler proofs of stronger results. Very general conformal measures are considered. Equivalent conditions are given for…
We study the regularity of Lyapunov exponents as functions on the space of compactly supported probability measures on $\mathrm{GL}(d,\mathbb{R})$. We prove that the Lyapunov exponents are pointwise log-H\"older continuous with respect to…
We consider a certain infinite product of random $2 \times 2$ matrices appearing in the solution of some $1$ and $1+1$ dimensional disordered models in statistical mechanics, which depends on a parameter $\varepsilon>0$ and on a real random…
I present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products $\Pi_n=M_nM_{n-1}\cdots M_1$, where $M_i$'s are i.i.d.. Following Tutubalin [Theor. Probab. Appl. {\bf 10},…
The Lyapunov exponents of a chaotic system quantify the exponential divergence of initially nearby trajectories. For Hamiltonian systems the exponents are related to the eigenvalues of a symplectic matrix. We make use of this fact to…
This paper is concerned with the study of random (Bernoulli and Markovian) product of matrices on a compact space of symbols. We establish the analyticity of the maximal Lyapunov exponent as a function of the transition probabilities, thus…
We construct a continuous linear cocycle over an expanding base dynamics for which the Lyapunov exponents of all ergodic invariant probability measures are small, except for one measure whose Lyapunov exponents are away from zero. The…
Linear skew-product semidynamical systems generated by random systems of delay differential equations are considered, both on a space of continuous functions as~well as on a space of $p$-summable functions. The main result states that in…
This paper is concerned with the Lyapunov spectrum for measurable cocycles over an ergodic pmp system taking values in semi-simple real Lie groups. We prove simplicity of the Lyapunov spectrum and its continuity under certain perturbations…
We consider a smooth expanding map g on the circle of degree 2. It is known that the Lyapunov exponent of g with respect to the unique invariant measure that is absolutely continuous with respect to the Lebesgue measure is positive and less…
Anomalies are known to appear in the perturbation theory for the one-dimensional Anderson model. A systematic approach to anomalies at critical points of products of random matrices is developed, classifying and analysing their possible…
Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external…
We prove an ergodic theorem for the rotation number of the composition of a sequence os stationary random homeomorphisms in $S^{1}$. In particular, the concept of rotation number of a matrix $g\in Gl^{+}(2,{\R})$ can be generalized to a…