Related papers: Random matrix products when the top Lyapunov expon…

We prove that, if $R$ is a non-discrete irreducible, continuous ring, then its unit group $\mathrm{GL}(R)$, equipped with the topology generated by the rank metric, is topologically simple modulo its center, path-connected, locally…

Let $G$ be a real connected algebraic semi-simple Lie group, and $H$ an algebraic subgroup of $G$. Let $\mu$ be a probability measure on $G$, with finite exponential moment, whose support spans a Zariski-dense subsemigroup of $G$. Let…

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…

Let $(g_n)_{n\geq 1}$ be a sequence of independent and identically distributed elements of the general linear group $GL(d, \mathbb R)$. Consider the random walk $G_n: = g_n \ldots g_1$. Under suitable conditions, we establish…

We study the linearization of a discrete transportation distance between probability distributions on finite weighted graphs originally due to Maas (``Gradient flows of the entropy for finite {M}arkov chains,'' J. Funct. Anal. 261(8), 2011)…

In this paper we study random walks on a finitely generated group $G$ which has a free action on a $\mathbb{Z}^n$-tree. We show that if $G$ is non-abelian and acts minimally, freely and without inversions on a locally finite…

We establish the universality of the singular numbers in random matrix products over $\mathrm{GL}_n(\mathbb{Q}_p)$ as the number of products approaches infinity, with a fixed $n\ge 1$. We demonstrate that, under a broad class of…

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,…

We study random walks on $\mathrm{GL}_d(\mathbb{R})$ whose proximal dimension $r$ is larger than $1$ and whose limit set in the Grassmannian $\mathrm{Gr}_{r,d}(\mathbb{R})$ is not contained any Schubert variety. These random walks, without…

Let G be a free product of a finite family of finite groups, with the set of generators being formed by the union of the finite groups. We consider a transient nearest-neighbour random walk on G. We give a new proof of the fact that the…

The top Lyapunov exponent $\lambda_+(A, p)$ of a random product of matrices in $\mathrm{GL}(d, \mathbb{R})$, $d \geq 2$, with simple top spectrum, depends real-analytically on the probability weights $p$ and the matrix coefficients $A$. We…

We study the Hopf property for wreath products of finitely generated groups, focusing on the case of an abelian base group. Our main result establishes a strong connection between this problem and Kaplansky's stable finiteness conjecture.…

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…

Let $(g_n)_{n\geq 1}$ be a sequence of independent and identically distributed random elements with law $\mu$ on the general linear group $\textrm{GL}(V)$, where $V=\mathbb R^d$. Consider the random walk $G_n : = g_n \ldots g_1$, $n \geq…

Products of $M$ i.i.d. random matrices of size $N \times N$ are related to classical limit theorems in probability theory ($N=1$ and large $M$), to Lyapunov exponents in dynamical systems (finite $N$ and large $M$), and to universality in…

The paper analyzes a specific class of random walks on quotients of $X:=\text{SL}(k,{\Bbb R})/ \Gamma$ for a lattice $\Gamma$. Consider a one parameter diagonal subgroup, $\{g_t\}$, with an associated abelian expanding horosphere, $U\cong…

Let $G$ be a countably infinite group, and let $\mu$ be a generating probability measure on $G$. We study the space of $\mu$-stationary Borel probability measures on a topological $G$ space, and in particular on $Z^G$, where $Z$ is any…

Let $V$ be a finite-dimensional unitary representation of a compact quantum group $\mathrm{G}$ and denote by $\mathrm{G}_W$ the isotropy subgroup of a linear subspace $W\le V$ regarded as a point in the Grassmannian $\mathbb{G}(V)$. We show…

We show that a common Lyapunov matrix exists for the convex combination of two Hurwitz matrices if and only if the intersection of the set of strict Lyapunov matrices for one matrix and the set of non-strict Lyapunov matrices for the other…

We show that the product or convex combination of two Markov operators with equivalent stationary measures need not have a stationary measure from the same measure class. More specifically, we exhibit examples of a hitherto undescribed…