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

Formulae for the Derivative of the Poincar\'e Constant of Gibbs Measures