This is a problem that has come up in my research and I am unsure of how approach it. Any help would be greatly appreciated.
Let $\pi^{\beta}$ be a probability distribution supported on $\mathbb{R}$ with bounded unnormalized density $\pi^{\beta}(x)=\gamma(x)^{\beta}/Z(\beta)$ for inverse-temperature $\beta\in (0,1]$. Let us define the energy as $E(x)=-\log(\gamma(x))$. If $X,Y\sim \pi^{\beta}$ are independent, then I am interested in the expected change in energy of the system at temperature $\beta$ as $\beta\to 0$. More precisely
Does $$\mathbb{E}_{\beta}(|E(X)-E(Y)|)=\int_{\mathbb{R}^2} |E(x)-E(y)|\pi^\beta(x)\pi^\beta(y)dxdy$$ diverge as $\beta\to0$?
I want to say yes, as all of the simulations I have done dirvege as $\beta\to 0$. In the case of the guasian, I compute the above expectation to be precisely $\frac{1}{\pi\beta}$.
We have,
$$\int_{\mathbb{R}^2} |E(x)-E(y)|\pi^\beta(x)\pi^\beta(y)dxdy =\int_{\mathbb{R}^2} |\log(\gamma(x)/\gamma(y))|\frac{\gamma^\beta(x)}{Z(\beta)}\frac{\gamma^\beta(y)}{Z(\beta)}dxdy$$ $$
Since $\pi^\beta$ is supported on $\mathbb{R}$ and $\gamma(x)^\beta\to 1$ pointwise, so we have $Z(\beta)\to\infty$ and $\pi^\beta([-K,K])\to 0$ for all $K>0$. So this leads me to think this is a tail issue. Any ideas on what I can do or possible conditions I need to impose on $\gamma$? Thank you in advanced!