The extremal position of a branching random walk on the general linear group

Abstract

Consider a branching random walk $(G_u{)}_{u\in \mathbb{T}}$ on the general linear group $\text{GL}(V)$ of a finite dimensional space $V$, where $\mathbb{T}$ is the associated genealogical tree with nodes $u$. For any starting point $v\in V\setminus \{0\}$ with $\Vert v\Vert =1$ and $x=\mathbb{R}v\in \mathbb{P}(V)$, let $M_n^x={\max }_{|u|=n}\log \Vert G_{u}v\Vert$ denote the maximal position of the walk $\log \Vert G_{u}v\Vert$ in the generation $n$. We first show that under suitable conditions, ${\lim }_{n\to \infty }\frac{M_n^x}{n}=\gamma$ almost surely, where $\gamma \in \mathbb{R}$ is a constant. Then, in the case when $\gamma =0$, under appropriate {\textit boundary conditions}, we refine the last statement by determining the rate of convergence at which $M_n^x$ converges to $-\infty$. We prove in particular that ${\lim }_{n\to \infty }\frac{M_n^x}{\log n}=-\frac{3}{2\alpha }$ in probability, where $\alpha >0$ is a constant determined by the boundary conditions. Analogous properties are established for the minimal position. As a consequence we derive the asymptotic speed of the maximal and minimal positions for the coefficients, the operator norm and the spectral radius of $G_u$.