Salez 与 Youssef 猜想的一个反例
A counterexample to a conjecture by Salez and Youssef
AI总结：
本文通过构造长度递增的生死链反例，否定了 Salez 和 Youssef 关于用 Ollivier 曲率替代 Bakry-Émery 曲率仍可下界 log-Sobolev 常数的猜想。
摘要：
近年来，在离散 Ricci 曲率有下界的假设下建立 log-Sobolev 型不等式取得了显著进展。更具体地说，Salez 和 Youssef 证明了 log-Sobolev 常数可以由 Bakry-Émery 曲率下界除以稀疏参数的对数来下界。他们猜想，当用 Ollivier 曲率替代 Bakry-Émery 曲率时，同样的结论仍然成立，而 Ollivier 曲率在实践中通常更容易计算。在本文中，我们通过在长度不断增加的生死链上给出一个反例，证明该猜想是错误的。
英文摘要：
Remarkable progress has been made in recent years to establish log-Sobolev type inequalities under the assumption of discrete Ricci curvature bounds. More specfically, Salez and Youssef have proven that the log-Sobolev constant can be lower bounded by the Bakry Emery curvature lower bound divided by the logarithm of the sparsity parameter. They conjectured that the same holds true when replacing Bakry Emery by Ollivier curvature which is often times easier to compute in practice. In this paper, we show that this conjecture is wrong by giving a counter example on birth death chains of increasing length.
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