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

Consider a branching random walk $(G_u)_{u\in \mathbb T}$ on the general linear group $\textrm{GL}(V)$ 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 $\|v\|=1$ and $x = \mathbb R v \in \mathbb P(V)$, let $M^x_n=\max_{|u| = n} \log \| G_u v \|$ denote the maximal position of the walk $\log \| G_u v \|$ 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 $G_u$.