Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution

Abstract

In this paper, consider the circular Cauchy distribution $\mu_x$ on the unit circle $S$ with index $0\le |x|<1$, we study the spectral gap and the optimal logarithmic Sobolev constant for $\mu_x$, denoted respectively as $\lambda_1(\mu_x)$ and $C_{\mathrm{LS}}(\mu_x).$ We prove that $\frac{1}{1+|x|}\le \lambda_1(\mu_x)\le 1$ while $C_{\mathrm{LS}}(\mu_x)$ behaves like $\log(1+\frac{1}{1-|x|})$ as $|x|\to 1.$