相关论文: Exact Lyapunov Exponent for Infinite Products of R…
The theory of products of random matrices and Lyapunov exponents have been widely studied and applied in the fields of biology, dynamical systems, economics, engineering and statistical physics. We consider the product of an i.i.d. sequence…
We study the top Lyapunov exponents of random products of positive $2 \times 2$ matrices and obtain an efficient algorithm for its computation. As in the earlier work of Pollicott, the algorithm is based on the Fredholm theory of…
The exact value of the Lyapunov exponents for the random matrix product $P_N = A_N A_{N-1}...A_1$ with each $A_i = \Sigma^{1/2} G_i^{\rm c}$, where $\Sigma$ is a fixed $d \times d$ positive definite matrix and $G_i^{\rm c}$ a $d \times d$…
We first study i.i.d. products of finitely many invertible $2 \times 2$ matrices with positive entries, and prove that the top Lyapunov exponent admits an explicit, rapidly convergent Neumann-series-type representation involving an infinite…
We consider a finite family of invertible $2 \times 2$ real matrices and a transitive Markov shift on the index set. Let $\lambda$ be the top Lyapunov exponent for random matrix products driven by the Markov shift. We prove that, if the…
The problems that we consider in this paper are as follows. Let $A_1, \ldots, A_k$ be square matrices (over reals). Let $W=w(A_1, \ldots, A_k)$ be a random product of $n$ matrices. What is the expected absolute value of the largest (in the…
Statistical properties of infinite products of random isotropically distributed matrices are investigated. Both for continuous processes with finite correlation time and discrete sequences of independent matrices, a formalism that allows to…
We study products of arbitrary random real $2 \times 2$ matrices that are close to the identity matrix. Using the Iwasawa decomposition of $\text{SL}(2,{\mathbb R})$, we identify a continuum regime where the mean values and the covariances…
We give lower and upper bounds on both the Lyapunov exponent and generalised Lyapunov exponents for the random product of positive and negative shear matrices. These types of random products arise in applications such as fluid stirring…
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…
The problems that we consider in this paper are as follows. Let A and B be 2x2 matrices (over reals). Let w(A, B) be a word of length n. After evaluating w(A, B) as a product of matrices, we get a 2x2 matrix, call it W. What is the largest…
We construct explicit invariant measures for a family of infinite products of random, independent, identically-distributed elements of SL(2,C). The matrices in the product are such that one entry is gamma-distributed along a ray in the…
Lyapunov exponents describe the asymptotic behavior of the singular values of large products of random matrices. A direct computation of these exponents is however often infeasible. By establishing a link between Lyapunov exponents and an…
We show that Lyapunov exponents and stability exponents are equal in the case of product of $i.i.d$ isotropic(also known as bi-unitarily invariant) random matrices. We also derive aysmptotic distribution of singular values and eigenvalues…
The paper provides a new integral formula for the largest Lyapunov exponent of Gaussian matrices, which is valid in the real, complex and quaternion-valued cases. This formula is applied to derive asymptotic expressions for the largest…
We analyze the top Lyapunov exponent of the product of sequences of two by two matrices that appears in the analysis of several statistical mechanics models with disorder: for example these matrices are the transfer matrices for the nearest…
The statistical behaviour of a product of independent, identically distributed random matrices in $\text{SL}(2,{\mathbb R})$ is encoded in the generalised Lyapunov exponent $\Lambda$; this is a function whose value at the complex number $2…
We establish large deviation type estimates for i.i.d. products of two dimensional random matrices with finitely supported probability distribution. The estimates are stable under perturbations and require no irreducibility assumptions. In…
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…
A random matrix with rows distributed as a function of their length is said to be isotropic. When these distributions are Gaussian, beta type I, or beta type II, previous work has, from the viewpoint of integral geometry, obtained the…