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

TL;DR

This paper studies the extremal behavior of a branching random walk on the general linear group, establishing almost sure limits and convergence rates for maximal and minimal positions, with implications for spectral properties.

Contribution

It introduces new results on the asymptotic behavior and convergence rates of extremal positions in a branching random walk on ext{GL}(V), including boundary condition effects.

Findings

Almost sure convergence of normalized maximal position to a constant 

Logarithmic convergence rate of maximal position to /2 in probability when =0

Asymptotic speeds for spectral radius and operator norm of the group elements

Abstract

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 $G_u$.