论文标题
在均匀多项式生长的平面图上
On planar graphs of uniform polynomial growth
论文作者
论文摘要
考虑一个无限的平面图，具有均匀的多项式生长d> 2。此类图的许多示例具有相似的几何和光谱特性，并且已经猜想这是必要的。我们提出了一个反例。特别是，我们表明，对于每个有理理性的d> 2，都有一个平面图，具有均匀的多项式生长D，随机步行是短暂的，反驳了Benjamini（2011）的猜想。 通过本杰米尼和施拉姆的众所周知的定理，这样的图不能是单模型的随机图。我们还提供了具有意外特性的均匀多项式生长的单模型随机平面图的示例。例如，（几乎确定）每个有理学度> 2的均匀多项式生长的图形图，步行的速度指数大于1/d，并且将所有球的补充都连接起来。这对本杰米尼和帕帕索格洛（2011）的两个问题进行了负面解决。
Consider an infinite planar graph with uniform polynomial growth of degree d > 2. Many examples of such graphs exhibit similar geometric and spectral properties, and it has been conjectured that this is necessary. We present a family of counterexamples. In particular, we show that for every rational d > 2, there is a planar graph with uniform polynomial growth of degree d on which the random walk is transient, disproving a conjecture of Benjamini (2011). By a well-known theorem of Benjamini and Schramm, such a graph cannot be a unimodular random graph. We also give examples of unimodular random planar graphs of uniform polynomial growth with unexpected properties. For instance, graphs of (almost sure) uniform polynomial growth of every rational degree d > 2 for which the speed exponent of the walk is larger than 1/d, and in which the complements of all balls are connected. This resolves negatively two questions of Benjamini and Papasoglou (2011).
