论文标题
随机步行装饰的Galton-Watson树
Random walks on decorated Galton-Watson trees
论文作者
论文摘要
在本文中，我们研究了一棵装饰有装饰的Galton-Watson树上的简单随机步行，该树从Galton-Watson树中获得了$ n $的每个顶点，并用独立的图形$ g_n $替换了$ n $的每个顶点，并沿树结构插入了插入的图。我们假设存在常量$ d，r \ geq 1，v <\ infty $，以至于直径，有效的电阻和$ g_n $的有效阻力像$ n^{\ frac {\ frac {1} {d}} {d}}，n^{\ frac {\ frac {1} {1} {1} {r}} {r}}，n^v $ n^v $ as $ n fty $我们还假设，基础的Galton-Watson树对于某些常数$ C> 0 $和（1,2）$的$ CX^{ - α} $如$ cx^{ - α} $的衰减至关重要。我们将所得度量空间的分形维度，光谱维度，步行维度和简单的随机步行位移指数作为$α，d，r $和$ v $的函数，以及这些数量波动的界限。
In this article, we study a simple random walk on a decorated Galton-Watson tree, obtained from a Galton-Watson tree by replacing each vertex of degree $n$ with an independent copy of a graph $G_n$ and gluing the inserted graphs along the tree structure. We assume that there exist constants $d, R \geq 1, v < \infty$ such that the diameter, effective resistance across and volume of $G_n$ respectively grow like $n^{\frac{1}{d}}, n^{\frac{1}{R}}, n^v$ as $n \to \infty$. We also assume that the underlying Galton-Watson tree is critical with offspring tails decaying like $cx^{-α}$ for some constant $c>0$ and some $α\in (1,2)$. We establish the fractal dimension, spectral dimension, walk dimension and simple random walk displacement exponent for the resulting metric space as functions of $α, d, R$ and $v$, along with bounds on the fluctuations of these quantities.
