We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. To establish this, we work directly at the level of Benjamini-Schramm limits, and exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and spectral expansion are incompatible "at infinity". We then transfer this result to finite graphs via local weak convergence. The same approach also applies to the Bacry-Emery curvature condition CD$(0,\infty)$, thereby settling a recent conjecture of Cushing, Liu and Peyerimhoff (2019).

翻译：我们证明,没有带有非阴性奥利维埃-里基曲曲线的封闭度扩张器,从而解决了Naor和Milman提出并由Olivier(2010年)公布的长期开放问题。事实上,即使我们允许大度、大电子值和负曲线边缘的消失比例,情况依然如此。为了确立这一点,我们直接在Benjamini-Schramm极限水平上工作,利用固定随机图对Liouville财产进行昆虫特征描述,以显示非阴性曲线和光谱扩展是“无限”不相容的。我们随后通过本地微弱的趋同将这一结果转移到有限的图表中。同样的做法也适用于培利-Emery曲线状况 CD$(0,\inty),从而解决了Cushing、Li和Pyerimihoff(2019年)最近的一个插图。