We introduce a strengthening of the notion of transience for planar maps in order to relax the standard condition of bounded degree appearing in various results, in particular, the existence of Dirichlet harmonic function s proved by Benjamini and Schramm. As a corollary we obtain that every planar nonamenable graph admits nonconstant Dirichlet harmonic function s27.

中文翻译：

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具有Liouville属性的每个平面图都是可以接受的

我们引入了对平面图的瞬态概念的增强，以放宽出现在各种结果中的有界度的标准条件，尤其是本杰米尼和施拉姆证明了Dirichlet调和函数的存在。作为推论，我们得到了每个平面不服从图都接受非恒定Dirichlet谐波函数s27。