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 prove 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 and a relative compactness argument. We believe that this "local weak limit" approach to mixing properties of Markov chains will have many other applications.

中文翻译：

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稀疏膨胀器的Ollivier-Ricci曲率为负

我们证明不存在具有非负Ollivier-Ricci曲率的有界数展开器，从而解决了Naor和Milman建议并由Ollivier（2010）公开的长期存在的开放问题。实际上，即使我们允许消失大比例，大特征值和负弯曲边缘的比例，也仍然如此。为了证明这一点，我们直接在Benjamini-Schramm极限水平上进行工作，并利用平稳随机图上Liouville属性的熵表征来显示非负曲率和频谱扩展在“无限远”不相容。然后，我们通过局部弱收敛和相对紧致度参数将此结果传输到有限图。我们相信，这种“局部弱极限”混合马尔可夫链特性的方法将有许多其他应用。