论文标题
泊松加权无限树上的有线最小跨越森林
The wired minimal spanning forest on the Poisson-weighted infinite tree
论文作者
论文摘要
我们研究有线最小跨越森林（WMSF）在泊松加权无限树（PWIT）上的光谱和扩散特性。令$ m $为PWIT上WMSF中包含根的树，而（y_n）_ {n \ geq0} $是从根部开始的$ m $上的简单随机步行。我们表明，几乎可以肯定的是，$ m $具有$ \ mathbb {p} [y_ {y_ {2n} = y_0] = n^{ - 3/4+o（1）} $和$ \ mathrm {dist}（y_0，y_n，y_n）= n^{1/4+o（y_n）= N^{1/4+o（o（o（o（o））。也就是说，$ m $的光谱尺寸为$ \ frac {3} {2} $，其典型的位移指数是$ \ frac {1} {4} {4} $，几乎可以肯定。这些证实了Addario-berry在ARXIV中的预测：1301.1667。
We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let $M$ be the tree containing the root in the WMSF on the PWIT and $(Y_n)_{n\geq0}$ be a simple random walk on $M$ starting from the root. We show that almost surely $M$ has $\mathbb{P}[Y_{2n}=Y_0]=n^{-3/4+o(1)}$ and $\mathrm{dist}(Y_0,Y_n)=n^{1/4+o(1)}$ with high probability. That is, the spectral dimension of $M$ is $\frac{3}{2}$ and its typical displacement exponent is $\frac{1}{4}$, almost surely. These confirm Addario-Berry's predictions in arXiv:1301.1667.
