相关论文: Lyapunov exponents for products of complex Gaussia…
For products $P_N$ of $N$ random matrices of size $d \times d$, there is a natural notion of finite $N$ Lyapunov exponents $\{\mu_i\}_{i=1}^d$. In the case of standard Gaussian random matrices with real, complex or real quaternion elements,…
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…
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…
We prove that the Lyapunov exponents of random products in a (real or complex) matrix group depends continuously on the matrix coefficients and probability weights. More generally, the Lyapunov exponents of the random product defined by any…
In this work, we give a rigorous explicit formula for the Lyapunov exponent for some binary infinite products of random $2\times 2$ real matrices. All these products are constructed using only two types of matrices, $A$ and $B$, which are…
Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random $N\times N$ matrices also called Ginibre ensemble we rederive the Lyapunov exponents for an infinite…
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 discuss several techniques for the evaluation of the generalised Lyapunov exponents which characterise the growth of products of random matrices in the large-deviation regime. A Monte Carlo algorithm that performs importance sampling…
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…
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…
Products of random $2\times 2$ matrices exhibit Gaussian fluctuations around almost surely convergent Lyapunov exponents. In this paper, the distribution of the random matrices is supported by a small neighborhood of order $\lambda>0$ of…
This article concerns the non-asymptotic analysis of the singular values (and Lyapunov exponents) of Gaussian matrix products in the regime where $N,$ the number of term in the product, is large and $n,$ the size of the matrices, may be…
This article gives a non-asymptotic analysis of the largest Lyapunov exponent of truncated orthogonal matrix products. We prove that as long as N, the number of terms in product, is sufficiently large, the largest Lyapunov exponent is…
Let $X_1,X_2, \ldots $ be a sequence of $i.i.d$ real (complex) $d \times d $ invertible random matrices with common distribution $\mu$ and $\sigma_1(n), \sigma_2(n), \ldots , \sigma_d(n)$ be the singular values, $\lambda_1(n), \lambda_2(n),…
For $m$ given square matrices $A_0, A_1, \cdots, A_{m-1}$ ($m\ge 2$), one of which is assumed to be of rank $1$, and for a given sequence $(\omega_n)$ in $\{0,1, \cdots, m-1\}^\mathbb{N}$, the following limit, if it exists,…
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 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 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…
Products of random matrix products of $\mathrm{SL}(2,\mathbb{R})$, corresponding to transfer matrices for the one-dimensional Schr\"odinger equation with a random potential $V$, are studied. I consider both the case where the potential has…
If A_1,...,A_N are real square matrices then the p-radius, generalised Lyapunov exponent or matrix pressure is defined to be the asymptotic exponential growth rate of the sum $\sum_{i_1,\ldots,i_n=1}^N \|A_{i_n}\cdots A_{i_1}\|^p$, where p…