We establish formulae for the derivative of the Poincaré constant of Gibbs measures on both compact domains and all of $\mathbb{R}$ ${}^d$. As an application, we show that if the (not necessarily convex) Hamiltonian is an increasing function, then the Poincaré constant is strictly decreasing in the inverse temperature, and vice versa. Applying this result to the $O\left(2\right)$ model allows us to give a sharpened upper bound on its Poincaré constant. We further show that this model exhibits a qualitatively different zero-temperature behavior of the Poincaré and Log-Sobolev constants.

中文翻译：

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吉布斯测度的庞加莱常数的导数公式

我们建立了吉布斯测度的庞加莱常数的导数公式，用于紧致域和所有 $\mathbb{电阻}$ ${}^d$. 作为一个应用，我们表明如果（不一定是凸的）哈密顿量是一个递增函数，那么庞加莱常数在逆温度中严格递减，反之亦然。将此结果应用于$哦\left(2\right)$模型允许我们给出其庞加莱常数的锐化上限。我们进一步表明，该模型展示了 Poincaré 和 Log-Sobolev 常数在性质上不同的零温度行为。