论文标题
一般线性组上分支随机步行的极端位置
The extremal position of a branching random walk on the general linear group
论文作者
论文摘要
考虑一个有限尺寸空间$ v $的一般线性组$ \ textrm {gl}（v）$上的分支随机步行$（g_u）_ {u \ in \ mathbb t} $，其中$ \ mthbb t $是与nodes $ u $相关的家谱。对于任何起点$ v \ in v \ setMinus \ {0 \} $带有$ \ | v \ | = 1 $和$ x = \ mathbb r v \ in \ mathbb p（v）$，让$ m^x_n = \ max_ {| max_ {| u | = n} \ log \ | g_u v \ | $表示步行$ \ log \ |的最大位置$ n $中的g_u v \ | $。我们首先表明，在适当的条件下，$ \ lim_ {n \ to \ infty} \ frac {m_n^x} {n} =γ$几乎肯定，其中$γ\ in \ mathbb r $是一个常数。然后，如果$γ= 0 $，在适当的{\ textit边界条件下}时，我们通过确定$ m_n^x $收敛到$ - \ infty $的收敛速率来完善最后的语句。我们特别证明了$ \ lim_ {n \ to \ infty} \ frac {m_n^x} {\ log n} = - \ frac {3} {2α} $，其中$α> 0 $是由边界条件确定的常数。建立了最小位置的类似特性。结果，我们得出了系数，操作员规范和$ g_u $的光谱半径的最大和最小位置的渐近速度。
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 $G_u$.
