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 (Coarse Geometry and Randomness, Volume 2100 of Lecture Notes in Mathematics. Springer, Cham, 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 (Proc Am Math Soc 139(11):4105–4111, 2011).

考虑具有阶数\（d> 2 \）的均匀多项式增长的无限平面图。这样的图的许多示例表现出相似的几何和光谱特性，并且已经推测这是必要的。我们提出了一系列反例。特别是，我们表明，对于每个有理\（d> 2 \），都有一个平面图，该图的次数为d的多项式均值增长随机游走是短暂的，这反驳了Benjamini的猜想（粗略的几何和随机性，《数学讲义》第2100卷，Springer，Cham，2011年）。根据本杰明（Benjamini）和施拉姆（Schramm）的著名定理，这样的图不能是单模随机图。我们还给出了具有意外特性的均匀多项式增长的单模随机平面图的示例。例如，每步有理数\（d> 2 \）的（几乎确定）均匀多项式增长的图形，其步速指数大于1 / d，并且所有球的补数都连接在一起。这消极地解决了Benjamini和Papasoglou的两个问题（Proc Am Math Soc 139（11）：4105-4111，2011）。