Emergence of the renormalized Gibbs measure

Let $d=2,3$, $h=-\Delta+\kappa>0$, and $0\leqslant \widehat w\in \ell^1(2\pi\mathbb{Z}^d)$. Let $\nu$ be as in the definition of the adjusted chemical potential, and consider the Gibbs state $\Gamma_\lambda=\mathcal{Z}_\lambda e^{-\mathbb{H}_\lambda/T}$ with

$$\mathbb{H}_\lambda=\int_{\Omega}a_x^*(-\Delta-\nu)a_x\,dx+\frac{\lambda}{2}\iint_{\Omega\times\Omega}a_x^*a_y^*w(x-y)a_xa_y\,dx\,dy.$$

Emergence of the renormalized Gibbs measure. In the limit $\lambda=T^{-1}\to0$, one obtains the renormalized Gibbs measure in the sense that

$$\operatorname{Tr}\left|\frac{k!}{T^k}\Gamma_\lambda^{(k)}-\int |u^{\otimes k}\rangle\langle u^{\otimes k}|\,d\mu(u)\right|^p\to0,$$

for every $k\geqslant1$ and every $p>d/2$. Here $\mu$ is the renormalized classical Gibbs measure associated with the preceding definition of the renormalized interaction. This conjectural quantum-classical correspondence predicts convergence of all reduced density matrices in the stated Schatten classes, but the supplied text does not indicate whether it has been proved or remains open.