The extremal position of a branching random walk on the general linear group
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929620783202304 |
|---|---|
| author | Grama, Ion Mentemeier, Sebastian Xiao, Hui |
| author_facet | Grama, Ion Mentemeier, Sebastian Xiao, Hui |
| contents | 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, $\lim_{n \to \infty} \frac{M_n^x }{n} = γ$ almost surely, where $γ\in \mathbb R$ is a constant. Then, in the case when $γ= 0$, 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 $\lim_{n \to \infty} \frac{M_n^x}{\log n} = -\frac{3}{2α}$ in probability, where $α>0$ 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 $G_u$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_04941 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The extremal position of a branching random walk on the general linear group Grama, Ion Mentemeier, Sebastian Xiao, Hui Probability Primary 60J80, Secondary 60B20, 60J05 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, $\lim_{n \to \infty} \frac{M_n^x }{n} = γ$ almost surely, where $γ\in \mathbb R$ is a constant. Then, in the case when $γ= 0$, 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 $\lim_{n \to \infty} \frac{M_n^x}{\log n} = -\frac{3}{2α}$ in probability, where $α>0$ 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 $G_u$. |
| title | The extremal position of a branching random walk on the general linear group |
| topic | Probability Primary 60J80, Secondary 60B20, 60J05 |
| url | https://arxiv.org/abs/2206.04941 |