\section{Introduction and main results}

We consider a lattice system of unbounded and continuous spins on the $d$-dimensional lattice $\mathds{Z}^d$. The formal Hamiltonian $H: \mathds{R}^{\mathds{Z}^d} \to \mathds{R}$ of the system is given by
\begin{equation}
  \label{e_d_Hamiltonian}
	H(x) = \sum_{i \in \mathds{Z}^d } \psi_i(x_i) + \frac{1}{2} \sum_{i,j \in \mathds{Z}^d} M_{ij} x_i x_j.
\end{equation}
We assume that the single-site potentials $\psi_i : \mathds{R} \to \mathds{R}$ are smooth and perturbed convex. This means that there is a splitting $\psi_i= \psi_i^c + \psi_i^b$ such that for all $i \in \mathds{Z}^d$ and $z \in \mathds{R}$
\begin{equation}\label{e_cond_psi}
  (\psi_i^c)'' (z) \geq 0 \qquad \mbox{and} \qquad |\psi_i^b (z)| + | ( \psi_i^b)' (z) | \lesssim 1.
\end{equation}
Here, we used the convention (see Definition~\ref{def:dep} below for more details)
\begin{equation*}
  a \lesssim b \qquad :\Leftrightarrow \mbox{there is a uniform constant $C>0$ such that $a \leq C b$}.
 \end{equation*}
Moreover, we assume that
\begin{itemize}
\item  the interaction is symmetric i.e.~
  \begin{equation}
    \label{e_ass_sym}
    M_{ij}=M_{ji} \qquad \mbox{ for all $i, j \in \mathds{Z}^d$,}
  \end{equation}

\item and the matrix $M= (M_{ij})$ is strictly diagonal dominant i.e.~for some $\delta > 0$ it holds for any $i \in \mathds{Z}^d$
\begin{equation}\label{e_strictly_diag_dominant}
  \sum_{j \in \mathds{Z}^d, j \neq i} |M_{ij}| + \delta \le M_{ii}.
\end{equation}
\end{itemize}

\begin{notation}
Let $S\subset \mathds{Z}^d$ be an arbitrary subset of $\mathds{Z}^d$. For convenience, we write $x^S$ as a shorthand for $(x_i)_{i \in S}$.
\end{notation}
\begin{definition}[Tempered spin-values]
Given a finite subset $\Lambda\subset \mathds{Z}^d$, we call the spin values $x^{\mathds{Z}^d \backslash \Lambda}$ tempered, if for all $i \in \Lambda$
\begin{equation*}
  \sum_{j \in \mathds{Z}^d \backslash {\Lambda}} |M_{ij}| \ |x_j| < \infty.
\end{equation*}
\end{definition}
\begin{definition}[Finite-volume Gibbs measure]
  Let $\Lambda$ be a finite subset of the lattice $\mathds{Z}^d$ and let $x^{\mathds{Z}^d \backslash \Lambda}$ be a tempered state. We call the measure $\mu_{\Lambda}( dx^{\Lambda})$ finite-volume Gibbs measure associated to the Hamiltonian $H$ with boundary values $x^{\mathds{Z}^d \backslash \Lambda}$, if it is a probability measure on the space $\mathds{R}^{\Lambda}$ given by the density
\begin{equation}
  \label{e_d_Gibbs_measure}
	\mu_{\Lambda}(dx^\Lambda) = \frac{1}{Z_{\mu_\Lambda}} \mathrm{e}^{-H(x^\Lambda,x^{\mathds{Z}^d \backslash \Lambda} )} \mathrm{d} x^\Lambda .
\end{equation}
Here, $Z_{\mu_\Lambda}$ denotes the normalization constant that turns $\mu_{\Lambda}$ into a probability measure. If there is no ambiguity, we also may write $Z$ to denote the normalization constant of a probability measure. We also used the short notation
\begin{align*}
  H(x^\Lambda,x^{\mathds{Z}^d \backslash \Lambda} ) = H(x) \quad \mbox{with} \quad x= (x^\Lambda,x^{\mathds{Z}^d \backslash \Lambda}).
\end{align*}
 Note that $\mu_\Lambda$ depends on the spin values $x^{\mathds{Z}^d \backslash \Lambda}$ outside of the set $\Lambda$.
\end{definition}

The main object of study in this article is the question if the finite-volume Gibbs measure $\mu_{\Lambda}$ satisfies a logarithmic Sobolev inequality~(LSI).
\begin{definition}[LSI] \label{d_LSI}
  Let $X$ be a Euclidean space. A Borel probability measure $\mu$ on $X$ satisfies the LSI with constant $\varrho>0$, if for all smooth functions $f \geq 0$
  \begin{equation}\label{e_definition_of_LSI}
  \int f \log f \ d \mu - \int f  d\mu  \log \left( \int f  d\mu \right) \leq \frac{1}{2 \varrho} \int \frac{|\nabla f|^2}{f} d\mu.
  \end{equation}
Here, $\nabla$ denotes the gradient determined by the Euclidean structure of~$X$.
\end{definition}
The LSI yields by linearization the Poincar\'e inequality (PI) (see for example~\cite{L}).
\begin{definition}[PI]\label{d_SG}
   Let $X$ be a Euclidean space. A Borel probability measure $\mu$ on $X$ satisfies the PI with constant $\varrho>0$, if for all smooth functions~$f$
\[
   \var_{\mu} (f): = \int \left(f - \int f d \mu \right)^2 d \mu \leq \frac{1}{\varrho} \int |\nabla f|^2 d\mu.
\]
Here, $\nabla$ denotes the gradient determined by the Euclidean structure of~$X$.
\end{definition}

The LSI was originally introduced by Gross \cite{Gross}. It can be used as a powerful tool for studying spin systems. The LSI implies exponential convergence to equilibrium of the naturally associated conservative diffusion process. The rate of convergence is given by the LSI constant $\varrho$ (cf.~\cite[Chapter~3.2]{Roy07}). At least in the case of finite-range interaction, independence from the system size of the LSI constant of the local Gibbs state directly yields the uniqueness of the infinite-volume Gibbs state (cf.~\cite{Roy07,Yos_2,Zitt}). \medskip

In the literature, there are several results known that connect the decay of spin-spin correlations to the validity of a LSI uniform in the system size \cite{StrZeg,StrZeg2,Zeg96,Yos99,Yos01,B-H1}. This means that a static property of the equilibrium state of the system is connected to a dynamic property namely the relaxation to the equilibrium. We refer the reader to the Section~2.2.~of the article of Otto \& Reznikoff~\cite{OR07}, which gives a nice overview and discussion on the results in the literature. Otto \& Reznikoff used the two-scale criterion for the LSI (cf.~\cite[Theorem~1]{OR07} or \cite[Theorem 3]{GORV}) to deduce the following statement:\medskip
\begin{theorem}[\mbox{\cite[Theorem 3]{OR07}}]\label{p_OR_original}
Consider the formal Hamiltonian $H:\mathds{R}^{\mathds{Z}^d} \to \mathds{R} $ given by~\eqref{e_d_Hamiltonian}. Assume that the single site potentials $\psi_i=\psi$ are given by a function of the form
\begin{align}
  \label{e_single_site_potential_otto}
  \psi(z) = \frac{1}{12} z^4 + \psi^b (z) \qquad \mbox{with} \quad | \frac{d^2}{dz^2} \psi^b (z)|\leq C.
\end{align}
Assume that the interaction is symmetric i.e. $M_{ij}= M_{ji}$ and has zero diagonal i.e. $M_{ii}=0$. Consider a subset~$\Lambda_{\mathrm{tot}} \subset \mathbb{Z}^d$. We assume the uniform control:
\begin{align}
  \label{e_decay_inter_Otto}
  |M_{ij}| \lesssim  \exp \left( - \frac{|i-j|}{C} \right)
  \end{align}
for $i,j \in \Lambda$ and
\begin{align}
  \label{e_decay_corr_Otto}
|  \cov_{\mu_{\Lambda}} (x_i,x_j)| \lesssim \exp \left( - \frac{|i-j|}{C} \right)
\end{align}
 uniformly in $\Lambda \subset \Lambda_{\mathrm{tot}}$ and $i,j \in \Lambda$. Here, $\mu_{\Lambda}$ denotes the finite-volume Gibbs measures $\mu_{\Lambda}$ given by~\eqref{e_d_Gibbs_measure}. \newline
Then the finite-volume Gibbs measure $\mu_{\Lambda_{\mathrm{tot}}}$ satisfies the LSI with constant $\varrho>0$ depending only on the constant $C>0$ in~\eqref{e_single_site_potential_otto}, \eqref{e_decay_inter_Otto}, and~\eqref{e_decay_corr_Otto}.
\end{theorem}

The most important feature of Theorem~\ref{p_OR_original} is that the LSI constant $\varrho$ is independent of the system size $|\Lambda_{\mathrm{tot}}|$ and of the spin values~$x^{\mathds{Z}^d\backslash \Lambda_{\mathrm{tot}}}$ outside of~$\Lambda_{\mathrm{tot}}$. The advantage of Theorem~\ref{p_OR_original} over existing results connecting a decay of correlations to a uniform LSI is that it can deal with infinite-range interaction (cf.~\cite{StrZeg,StrZeg2,Zeg96,Yos99,Yos01,B-H1}). However, Theorem~\ref{p_OR_original}  calls for some technical improvements. The main result of this article is the following generalized version of Theorem~\ref{p_OR_original}:
\begin{theorem}[Generalization of \mbox{\cite[Theorem 3]{OR07}}]\label{p_mr_OR}
 Assume that the formal Hamiltonian $H:\mathds{R}^{\mathds{Z}^d} \to \mathds{R} $ given by~\eqref{e_d_Hamiltonian} satisfies the Assumptions~\eqref{e_cond_psi}~-~\eqref{e_strictly_diag_dominant}. Let $\Lambda_{\mathrm{tot}} \subset \mathds{Z}^d$ be an arbitrary, finite subset of the lattice $\mathds{Z}^d$.\newline
Assume the following decay of interactions and correlations: For some $\alpha>0$ it holds
  \begin{equation}\label{e_cond_inter_alg_decay_OR}
  |M_{ij}| \lesssim \frac{1}{|i-j|^{d+ \alpha}}
  \end{equation}
 uniformly in $i,j \in \Lambda_{\mathrm{tot}}$ and
\begin{equation}\label{e_cond_alg_decay_OR}
 |  \cov_{\mu_{\Lambda}} (x_i,x_j)|  \lesssim \frac{1}{|i-j|^{d + \alpha}}
\end{equation}
 uniformly in $\Lambda \subset \Lambda_{\mathrm{tot}}$, and $i,j \in \Lambda$. Here, $\mu_{\Lambda}$ denote the finite-volume Gibbs measures given by (cf.~\eqref{e_d_Gibbs_measure}).\newline
Then the finite-volume Gibbs measure $\mu_{\Lambda_{\mathrm{tot}}}$ satisfies the LSI with a constant $\varrho>0$ depending only on the constant in~\eqref{e_cond_psi},~\eqref{e_strictly_diag_dominant}, \eqref{e_cond_alg_decay_OR} and \eqref{e_cond_inter_alg_decay_OR}.
\end{theorem}

Theorem~\ref{p_mr_OR} improves Theorem~\ref{p_OR_original} in two ways:\newline
Note that Theorem~\ref{p_OR_original} needs an exponential decay of interaction and spin-spin correlations. However, analyzing the proof of~\cite[Theorem~3]{OR07} one sees that the exponential decay is only needed to guarantee that certain sums are summable. Therefore this assumption can be weakened to algebraically decaying interaction and spin-spin correlations. Of course now, the order of the algebraic decay depends on the dimension of the underlying lattice to guarantee summability. \smallskip

The second improvement is more subtle. Theorem~\ref{p_OR_original} needs a special structure on the single-site potentials $\psi_i$. Namely, the single-site potentials $\psi_i$ have to be perturbed quartic in the sense of~\eqref{e_single_site_potential_otto}. Analyzing the proof of~\cite[Theorem~3]{OR07} shows that the argument does not rely on a quartic potential~$\psi_i^c$. For the argument of Otto \& Reznikoff it would be sufficient to have a perturbation of a strictly-superquadratic potential~i.e.
\begin{equation}
  \label{e_super_quadratic}
  \liminf_{|x|\to \infty} \frac{d^2}{dx^2} \psi_i^c (x) \to \infty.
\end{equation}
The condition~\eqref{e_super_quadratic} on the single-site potential $\psi_i$ is widespread and accepted in the literature on the uniform LSI (cf.~for example~\cite{Yos01,Yos_2,ProSco}). \smallskip

However, a result by Zegarlinski~\cite[Theorem~4.1.]{Zeg96} indicates that the condition~\eqref{e_super_quadratic} is not necessary for deducing a uniform LSI. Zegarlinski deduced in~\cite[Theorem~4.1.]{Zeg96} the uniform LSI for the finite-volume Gibbs measure $\mu_\Lambda$ given by~\eqref{e_d_Gibbs_measure} on an one-dimensional lattice $\Lambda_{\mathrm{tot}} \subset \mathds{Z}$ with finite-range interaction.  For Zegarlinski's argument it is sufficient that the single-site potentials $\psi_i$ satisfy the conditions~\eqref{e_cond_psi} and~\eqref{e_strictly_diag_dominant}, which is strictly weaker than the condition~\eqref{e_super_quadratic} (for a proof of this statement we refer the reader to \cite[Proof of Lemma~1]{OR07}). In Theorem~\ref{p_mr_OR} we show that the conditions~\eqref{e_cond_psi} and~\eqref{e_strictly_diag_dominant} are in fact also sufficient for the Otto-Reznikoff approach.

\begin{remark}\label{r_linear_term}
 Note that the structural assumptions~\eqref{e_cond_psi}~-~\eqref{e_strictly_diag_dominant} on the Hamiltonian $H$ are invariant under adding a linear term like
  \begin{equation*}
    \sum_{i \in \mathds{Z}} x_ib_i
  \end{equation*}
for arbitrary $b_i \in \mathds{R}$. Therefore the the LSI constant of Theorem~\ref{p_mr_OR} is invariant under adding a linear term to the Hamiltonian. Such a linear term can be interpreted as a field acting on the system. If the coefficients $b_i$ are chosen randomly, one calls the linear term random field.
\end{remark}

Let us discuss what are the ingredients to weaken the structural assumptions on the single-site potential from the condition~\eqref{e_single_site_potential_otto} to the condition~\eqref{e_cond_psi} and~\eqref{e_strictly_diag_dominant}. Analyzing the proof of Otto \& Reznikoff, it all boils down to understanding the structure of the Hamiltonian of the marginals of the finite-volume Gibbs measure conditioned on the spin values of some set $S \subset \Lambda_{\mathrm{tot}}$ (cf.~\cite[Lemma 2, Lemma 3 and Lemma 4]{OR07} or see Section~\ref{s_LSI}). Because our structural assumptions ~\eqref{e_cond_psi} and~\eqref{e_strictly_diag_dominant} on the single-site potentials are weaker, our proof needs new ingredients and more detailed arguments compared to~\cite{OR07}. \medskip

The first new ingredient in the proof of Theorem~\ref{p_mr_OR} is the covariance estimate of Proposition~\ref{p_algebraic_decay_correlations}. With this estimate it is possible to deduce algebraic decay of correlations, provided the interactions $M_{ij}$ also decay algebraically and the nonconvex perturbation~$\psi_i^b$ is small enough.\medskip

The second new ingredient in the proof of Theorem~\ref{p_mr_OR} is a uniform estimate of $\var_{\mu_{\Lambda}} (x_i)$ (see Lemma~\ref{p_est_var_ss}), which we reduce to a moment estimate due to Robin Nittka (cf.~\cite[Lemma 4.2]{MN} and Lemma~\ref{lem:moments}). The full proof of Theorem~\ref{p_mr_OR} is given in Section~\ref{s_LSI}.\medskip

However, Theorem~\ref{p_mr_OR} still calls for further improvements. Note that in the condition~\eqref{e_cond_alg_decay_OR} of Theorem~\ref{p_mr_OR} one needs to check the decay of correlations for all finite-volume Gibbs measures $\mu_\Lambda$ with $\Lambda \subset \Lambda_{\mathrm{tot}}$. Even if this is a very common assumption (see for example \cite[Condition (DS3)]{Yos01}) it may be a bit tedious to verify. Instead of the strong condition~\eqref{e_cond_alg_decay_OR}, one would like to have a weak condition like the one used for discrete spins in~\cite{MaOl}. The main difference between the weak and the strong condition for the decay of correlations is that in the weak condition it suffices to show that for a sufficiently large box $\Lambda$ the correlations decay nicely. The main advantage of the weak condition is that one does not have to control the decay of correlations for all growing subsets $\Lambda \to \mathds{Z}^d$. Therefore, the weak condition is easier to verify by experiments. Unfortunately, we cannot get rid of the strong decay of correlations condition~\eqref{e_cond_alg_decay_OR} in the Otto-Reznikoff approach. However, we show how verifying the strong decay of correlations condition~ \eqref{e_cond_alg_decay_OR} can be simplified by two comparison principles.\smallskip

The first comparison principle (see Lemma~\ref{p_comparison_covariances} below) shows that in the case of ferromagnetic interaction (i.e.~$M_{ij}<0$ for all $i,j \in \Lambda_{\mathrm{tot}}$) the correlations of a smaller system are controlled by correlations of the larger system.

\begin{lemma}\label{p_comparison_covariances}
Assume that the formal Hamiltonian $H:\mathds{R}^{\mathds{Z}^d} \to \mathds{R} $ given by~\eqref{e_d_Hamiltonian} satisfies the Assumptions~\eqref{e_cond_psi}~-~\eqref{e_strictly_diag_dominant}.  Additionally, assume that the interactions are ferromagnetic i.e.~$M_{i,j} \leq 0$ for $i \neq j$. \newline
 For arbitrary subsets $\Lambda \subset \Lambda_{\mathrm{tot}} \subset \mathds{Z}^d$, we consider the finite-volume Gibbs measure $\mu_{\Lambda}$ and $\mu_{\Lambda_{\mathrm{tot}}}$ with the same tempered state $x^{\mathds{Z}^d \backslash \Lambda_{\mathrm{tot}}}$. Then it holds for any $i,j \in \Lambda$
 \begin{align} \label{e_comp_small_big_corr}
 \cov_{\mu_{\Lambda}} (x_i,x_j) \leq \cov_{\mu_{\Lambda_{\mathrm{tot}}}}(x_i,x_j) .
 \end{align}
\end{lemma}
The proof of Lemma~\ref{p_comparison_covariances} is given in Section~\ref{s_covariance_estimate}. The second comparison principle is rather standard. It states that correlations of a non-ferromagnetic system are controlled by the correlations of the associated ferromagnetic system:

\begin{lemma}\label{p:attractive_interact_dominates}
Assume that the formal Hamiltonian $H:\mathds{R}^{\mathds{Z}^d} \to \mathds{R} $ given by~\eqref{e_d_Hamiltonian} satisfies the Assumptions~\eqref{e_cond_psi}~-~\eqref{e_strictly_diag_dominant}.
Let $\mu_{\Lambda}$ denote the finite-volume Gibbs measure given by~\eqref{e_d_Gibbs_measure}. Additionally, consider the corresponding finite-volume Gibbs measure~~$\mu_{\Lambda,|M|}$ with attractive interaction i.e.~the associated formal Hamiltonian is given by
\begin{equation*}

	H(x) = \sum_{i \in \mathds{Z}^d } \psi_i(x_i) - \frac{1}{2} \sum_{i,j \in \mathds{Z}^d} |M_{ij}| x_i x_j.
\end{equation*}
Then it holds that for any $i,j \in \Lambda$
\begin{equation}
  \label{eq:covariance_domination}
  | \cov_{\mu_{\Lambda}} (x_i,x_j)  | \leq \cov_{\mu_{\Lambda,|M|}} (x_i,x_j) .
\end{equation}
\end{lemma}
We do not state the proof of the last lemma. One can find the proof for example in a recent work by Robin Nittka and the author. The proof follows the argument of~\cite{HorMor} for discrete spins (see~\cite[Lemma 2.1.]{MN}).

\begin{remark}
Usually, one considers finite-volume Gibbs measures for some inverse temperature $\beta >0$ i.e.
  \begin{equation*}
    	\mu_\Lambda (d x^\Lambda ) = \frac{1}{Z_\mu} \mathrm{e}^{- \beta H(x^\Lambda, x^{\mathds{Z} \backslash \Lambda})} \mathrm{d} x \qquad \mbox{for } x^\Lambda \in \mathds{R}^{\Lambda}.
  \end{equation*}
This case is also contained in the main results of the article, because the Hamiltonian $\beta H$ still satisfies the structural Assumptions~\eqref{e_cond_psi}~-~\eqref{e_strictly_diag_dominant}. Of course, the LSI constant of Theorem~\ref{p_mr_OR} would depend on the inverse temperature~$\beta$.
\end{remark}

\begin{remark}
  Because we assume that the matrix $M= (M_{ij})$ is strictly diagonal dominant (cf.~\eqref{e_strictly_diag_dominant}), the full single-site potential
  \begin{equation*}
    \psi_i(x_i) + M_{ii} x_i^2 = M_{ii} x_i^2 + \psi_i^c(x_i) + \psi_i^b (x_i)
  \end{equation*}
is perturbed strictly-convex. We want to note that this is the same structural assumption as used in the article~\cite{MO}.
\end{remark}

Let us turn to an application of Theorem~\ref{p_mr_OR}. We will show how the decay of correlations condition~\eqref{e_cond_alg_decay_OR} combined with the uniform LSI of Theorem~\ref{p_mr_OR} yields the uniqueness of the infinite-volume Gibbs measure. The statement that a uniform LSI yields the uniqueness of the Gibbs state is already known from the case of finite-range interaction (cf. for example~\cite{Yos_2}, the~conditions (DS1), (DS2), and (DS3) in~\cite{Yos01}). The related arguments of~\cite{Roy07},~\cite{Zitt}, and~\cite{Yos01} are based on semigroup properties of an associated diffusion process. Though the semigroup probably may work in the case of infinite-range interaction, we follow a more straightforward approach to deduce the uniqueness of the Gibbs measure. Before we formulate the precise statement (see Theorem~\ref{p_unique_Gibbs} below), we specify the notion of an infinite-volume Gibbs measure.
\begin{definition}[Infinite-Volume Gibbs measure]
  Let $\mu$ be a probability measure on the state space $\mathds{R}^{\mathds{Z}^d}$ equipped with the standard product Borel sigma-algebra. For any finite subset $\Lambda \subset \mathds{Z}^d$ we decompose the measure $\mu$ into the conditional measure $\mu(dx^\Lambda| x^{\mathds{Z}^d \backslash \Lambda})$ and the marginal $\bar \mu (d x^{\mathds{Z}^d \backslash \Lambda})$. This means that for any test function $f$ it holds
\begin{equation*}
  \int f(x) \mu (dx) = \int \int f(x) \mu(dx^\Lambda| x^{\mathds{Z}^d \backslash \Lambda}) \bar \mu (d x^{\mathds{Z}^d \backslash \Lambda}).
\end{equation*}
We say that the measure $\mu$ is the infinite-volume Gibbs measure associated to the Hamiltonian $H$, if the conditional measures $\mu(dx^\Lambda| x^{\mathds{Z}^d \backslash \Lambda})$ are given by the finite-volume Gibbs measures $\mu_{\Lambda}(dx^\Lambda)$ defined by~\eqref{e_d_Gibbs_measure} i.e.
\begin{equation*}
  \mu(dx^\Lambda| x^{\mathds{Z}^d \backslash \Lambda}) = \mu_\Lambda (dx^\Lambda).
\end{equation*}
The equations of the last identity are also called Dobrushin-Lanford-Ruelle (DLR) equations.
\end{definition}
The precise statement connecting the decay of correlations with the uniqueness of the infinite-volume Gibbs measure is:
\begin{theorem}[Uniqueness of the infinite-volume Gibbs measure]\label{p_unique_Gibbs}
 Under the same assumptions as in Theorem~\ref{p_mr_OR}, there is at most one unique Gibbs measure $\mu$ associated to the Hamiltonian $H$ satisfying the uniform bound
\begin{equation}~\label{e_sup_moment}
    \sup_{i \in \mathds{Z}^d} \int (x_i)^2 \mu (dx) < \infty.
\end{equation}
 \end{theorem}
The moment condition~\eqref{e_sup_moment} in Theorem~\ref{p_unique_Gibbs} is standard in the study of infinite-volume Gibbs measures (see for example~\cite{BHK82} and~\cite[Chapter~4]{Roy07}). It is relatively easy to show that the condition~\eqref{e_sup_moment} is invariant under adding a bounded random field to the Hamiltonian~$H$ (cf.~Remark~\ref{r_linear_term}). \smallskip

Theorem~\ref{p_unique_Gibbs} is one of the \emph{well-known} statements for which it is hard to find a proof. Therefore we state the proof in full detail in the Appendix~\ref{s_decay_and_uniqueness}. The argument does not need that the finite-volume Gibbs measures $\mu_{\Lambda}$ satisfy a uniform LSI. It suffices that the finite-volume Gibbs measures $\mu_{\Lambda}$ satisfy a uniform PI, which is a weaker condition then the LSI (see~Definition~\ref{d_SG}).   \medskip

We also want to note that the main results of this article, namely Theorem~\ref{p_mr_OR} and Theorem~\ref{p_unique_Gibbs} were applied in~\cite{MN} to deduce a uniform LSI and the uniqueness of the infinite-volume Gibbs measure of a one-dimensional lattice system with long-range interaction, generalizing Zegarlinsk's result~\cite[Theorem~4.1.]{Zeg96} to interactions of infinite range.

\begin{remark}
  In this article, we do not show the existence of an infinite-volume Gibbs measure. However, the author of this article believes that under the assumption~\eqref{e_sup_moment} the existence should follow by an compactness argument similarly to the one used in~\cite{BHK82}.
\end{remark}

In order to avoid confusion, let us make the notation $a \lesssim b$ from above precise.
\begin{definition}\label{def:dep}
	We will use the notation $a \lesssim b$ for quantities $a$ and $b$
	to indicate that there is a constant $C \ge 0$
	which depends only on a lower bound for $\delta$ and upper bounds for $|\psi_i^b|$, $|(\psi_i^b)'|$, and $\sup_i \sum_{j \in \mathds{Z}^d} |M_{ij}|$ such that $a \le C b$. In the same manner, if we assert the existence of certain constants, they may freely depend on the above mentioned quantities, whereas all other dependencies will be pointed out.
\end{definition}

We close the introduction by giving an outline of the article.\smallskip
\begin{itemize}
\item In Section~\ref{s_covariance_estimate}, we prove Lemma~\ref{p_comparison_covariances}. This contains the comparison principle for covariances of smaller systems to larger systems.
\item In Section~\ref{s_LSI}, we consider the generalization of Theorem~\ref{p_OR_original} and give the proof of Theorem~\ref{p_mr_OR}.
\item In the Appendix~\ref{s_decay_and_uniqueness}, we consider the uniqueness of the infinite-volume Gibbs measure and give the proof of Theorem~\ref{p_unique_Gibbs}.
\item In the Appendix~\ref{s_BE_HS} we state some well-known facts about the LSI and the PI.
\end{itemize}

\section{Comparing covariances of a smaller system to covariances to a bigger system: Proof of Lemma~\ref{p_comparison_covariances}}\label{s_covariance_estimate}

The proof of Lemma~\ref{p_comparison_covariances} uses an idea of Sylvester of expanding the exponential function~\cite{Sylvester}. Sylvester used this idea to give a simple unified derivation of a bunch of correlation inequalities for ferromagnets.

\begin{proof}[Proof of Lemma~\ref{p_comparison_covariances}]
  We fix the spin values $m_i$, $i \in \Lambda_{\mathrm{tot}} \backslash \Lambda$. Recall that in our notations $\mu_\Lambda$ coincides with the conditional measure
  \begin{align*}
    \mu_{\Lambda}(d x^{\Lambda}) = \mu_{\Lambda_{\mathrm{tot}}}(dx^\Lambda | m^{\Lambda_{\mathrm{tot}} \backslash \Lambda }).
  \end{align*}
We introduce the auxiliary Hamiltonian $H_\alpha$, $\alpha >0$, by the formula
  \begin{align*}
    H_\alpha (x) = H(x) + \alpha \sum_{i \in \Lambda_{\mathrm{tot}}\backslash \Lambda }(x_i - m_i)^2.
  \end{align*}
We denote by $\mu_{\alpha}$ the associated Gibbs measure active on the sites $\Lambda_{\mathrm{tot}}$. The measure $\mu_{\alpha}$ is given by the density
\begin{align*}
  \mu_{\alpha} (dx) = \frac{1}{Z} \ \exp \left( - H_{\alpha} (x) \right) \ d x \qquad \mbox{for } x \in \mathbb{R}^{\Lambda_{\mathrm{tot}}}.
\end{align*}
Note that the measure $\mu_{\alpha}$ interpolates between the measure $\mu_{\Lambda}$ and $\mu_{\Lambda_{\mathrm{tot}}}$ in the sense that $\mu_0 = \mu_{\Lambda_{\mathrm{tot}}}$ and for any integrable function $f: \mathbb{R}^{\Lambda} \to \mathbb{R}$
\begin{align*}
 \lim_{\alpha \to \infty} \int f(x^\Lambda)  \mu_\alpha (dx^{\Lambda_{\mathrm{tot}}}) = \int f(x^\Lambda) \mu(dx^\Lambda | m^{\Lambda_{\mathrm{tot}} \backslash \Lambda}).
\end{align*}
So we formally have $\mu_{\infty} = \mu_{\Lambda}$
Therefore it also holds for $i, j \in \Lambda$
\begin{align*}
   \lim_{\alpha \to \infty} \cov_{\mu_\alpha} (x_i, x_j) = \cov_{\mu_{\Lambda}} (x_i, x_j)
\end{align*}
This yields by the fundamental theorem of calculus that
\begin{align*}
  \cov_{\mu_\infty} (x_i, x_j) - \cov_{\mu_0} (x_i, x_j) = \int_0^\infty \frac{d}{d \alpha} \cov_{\mu_\alpha} (x_i, x_j).
  \end{align*}
We will now show that $$\frac{d}{d \alpha} \cov_{\mu_\alpha} (x_i, x_j) < 0,$$ which yields the statement of Lemma~\ref{p_comparison_covariances}.\newline
Indeed, direct calculation shows that
\begin{align*}
 & \frac{d}{d \alpha} \cov_{\mu_\alpha} (x_i, x_j)\\
 & \quad =   \frac{d}{d \alpha}  \left( \int x_i x_j \mu_{\alpha} - \int x_i \mu_{\alpha} \int x_j \mu_{\alpha}  \right) \\
  & \quad = - \cov_{\mu_\alpha} \left(  x_i x_j  - \int x_i \mu_{\alpha} \int x_j \mu_{\alpha} , \sum_{l \in \Lambda_{\mathrm{tot}} \backslash \Lambda }(x_l - m_l)^2 \right) \\
    & \quad = - \cov_{\mu_\alpha} \left(  x_i x_j , \sum_{l \in \Lambda_{\mathrm{tot}}\backslash \Lambda }(x_l - m_l)^2 \right).
\end{align*}
We will show now that
\begin{align}
  \cov_{\mu_\alpha} \left(  x_i x_j  , \sum_{l \in \Lambda_{\mathrm{tot}} \backslash \Lambda }(x_l - m_l)^2 \right) \geq 0. \label{e_cond_covariances_crucial_estimate}
\end{align}
For this purpose, we follow the method by Sylvester~\cite{Sylvester} of expanding the interaction term. Recall that this method is also used to show for example that
\begin{align*}
  \cov_{\mu_\alpha} \left(  x_i , x_j \right) \geq 0,
\end{align*}
provided the interactions are ferromagnetic. By doubling the variables we get
\begin{align*}
&    \cov_{\mu_\alpha} \left(  x_i x_j , \sum_{l \in \Lambda_{\mathrm{tot}}\backslash \Lambda }(x_l - m_l)^2 \right) \\
& = \int ( x_i x_j - \tilde x_i \tilde x_j ) \sum_{l \in \Lambda_{\mathrm{tot}} \backslash \Lambda }(x_l - m_l)^2 \mu_\alpha (dx) \ \mu_\alpha (d \tilde x) \\
&=  \frac{1}{Z^2}\int ( x_i x_j - \tilde x_i \tilde x_j ) \sum_{l \in \Lambda_{\mathrm{tot}} \backslash \Lambda }(x_l - m_l)^2 \exp (- H_\alpha (x) - H_\alpha (\tilde x)) dx d \tilde x
\end{align*}
Because the partition function $Z>0$ is positive, the sign of the covariance is determined by the integral on the right hand side of the last identity.
We change variables according to $x_i=  (p_i + q_i)$ and $\tilde x_i =  (p_i - q_i)$ and get
\begin{align}
&  \int ( x_i x_j - \tilde x_i \tilde x_j ) \sum_{l \in \Lambda_{\mathrm{tot}\backslash \Lambda }}(x_l - m_l)^2 \exp (- H_\alpha (x) - H_\alpha (\tilde x)) dx d \tilde x \notag \\
& \quad  = C \int ( (p_i + q_i) (p_j + q_j) - (p_i - q_i) (p_j - q_j) ) \notag \\
& \qquad \times \sum_{l \in \Lambda_{\mathrm{tot}}\backslash \Lambda }((p_l + q_l) - m_l)^2 \exp (- H_\alpha (p-q) - H_\alpha (p+q)) dp dq \notag \\
& \quad = C \int ( 2 p_i  q_j  + 2 q_i p_j)  \notag \\
& \qquad \times \sum_{l \in \Lambda_{\mathrm{tot}}\backslash \Lambda }((p_l + q_l) - m_l)^2 \exp (- H_\alpha (p-q) - H_\alpha (p+q)) dp dq  \label{e_integral_to_expand}
\end{align}
where $C>0$ is the constant from the transformation. Straightforward calculation reveals
\begin{align}
&   \tilde H_\alpha (p,q)  = H_\alpha (p-q) + H_\alpha (p+q) \notag \\
 &  = \sum_l   \psi_l (p_l-q_l) + \psi_l (p_l + q_l) +4 m_l p_l  + \alpha (p_l -  q_l )^2 + \alpha (p_l + q_l )^2 + 2 m_l^2  \notag \\
 & \qquad \qquad + 2 p \cdot  M  p + 2 q \cdot M q  . \label{e_calc_tilde_H}
\end{align}
For convenience, we only consider the first summand on the right hand side of~\eqref{e_integral_to_expand}. The second summand can be estimated in the same way. \newline
Due to symmetry of $\tilde H_\alpha (p,q)$ in the $q_l$ variables it holds
\begin{align*}
\int   q_j  \sum_{l \in \Lambda_{\mathrm{tot}}\backslash \Lambda }((p_l + q_l) - m_l)^2 \exp (-   \tilde H_\alpha (p,q)) dp dq  = 0
\end{align*}
Therefore we get by doubling the variable $p$ first and then changing of variables $p = r + \tilde q$ and $\tilde p = r - \tilde q$ that
\begin{align}
& \int  2 p_i  q_j  \sum_{l \in \Lambda_{\mathrm{tot}\backslash \Lambda }}((p_l + q_l) - m_l)^2 \exp (- \tilde H_\alpha (p,q)) dp dq  \notag \\
& \quad = \frac{1}{Z} \int  2 (p_i - \tilde p_i )  q_j  \sum_{l \in \Lambda_{\mathrm{tot}\backslash \Lambda }}((p_l + q_l) - m_l)^2 \notag \\
& \qquad \qquad \times \exp (- \tilde H_\alpha (p,q)- \tilde H_\alpha (\tilde p,q) ) d \tilde p dp dq \notag \\
&  \quad = \frac{1}{Z} \int  4 \tilde q_i   q_j  \sum_{l \in \Lambda_{\mathrm{tot}\backslash \Lambda }}((r_l+\tilde q_l  + q_l) - m_l)^2 \notag \\
& \qquad \qquad \times \exp (-  \tilde{\tilde{H}}_\alpha (r,\tilde q,q) ) d \tilde q dq dr , \label{e_integral_to_expand_finally}
\end{align}
where the Hamiltonian $\tilde{\tilde{H}}_\alpha (r,\tilde q,q)$ is given by
\begin{align*}
&  \tilde{\tilde{H}}_\alpha (r,\tilde q,q) \\
& = \tilde H_\alpha (r + \tilde q , q) + \tilde H_\alpha (r - \tilde q , q) \\
& = H_\alpha (r + \tilde q-q) + H_\alpha (r + \tilde q+q) + H_\alpha (r - \tilde q-q) + H_\alpha (r - \tilde q+q)
\end{align*}
As we have seen in~\eqref{e_calc_tilde_H} from above, the Hamiltonian $\tilde{\tilde{H}}_\alpha (r,\tilde q,q)$ contains no mixed terms in the variables $r, \tilde q$ and $q$. More precisely, $\tilde{\tilde{H}}_\alpha (r,\tilde q,q)$ has three interaction terms i.e.
\begin{align*}
4  r \cdot M r, \qquad 4 \tilde q \cdot M \tilde q , \qquad \mbox{and} \qquad 2 q \cdot M q.
\end{align*}
So we can rewrite  $\tilde{\tilde{H}}_\alpha (r,\tilde q,q)$ as
\begin{align*}
   \tilde{\tilde{H}}_\alpha (r,\tilde q,q) & = F(r,\tilde q, q) + 4  r \cdot M r + 4  \tilde q \cdot M \tilde q + 2 q \cdot M q,
\end{align*}
where the function $F$ is of the form
\begin{align*}
  F(r, \tilde q, q) = \sum_l \tilde{\psi}_l (r_l, \tilde q_l , q_l)
\end{align*}
for some single-site potentials $\tilde{\psi}_l$ that are symmetric in the variables $\tilde q_l$ and $q_l$. Expanding the term
\begin{align*}
   \exp(  - 4 \tilde q \cdot M \tilde q  - 2 q \cdot M q)
\end{align*}
on the right hand side of~\eqref{e_integral_to_expand_finally} yields a sum of terms of the form
\begin{align*}
& -M_{mn}  \int  q_{l_1}^{n_1} \cdots q_{l_k}^{n_k} \tilde q_{\tilde l_1}^{\tilde n_1} \ldots \tilde q_{\tilde l_1}^{\tilde n_1}  ((r_l+\tilde q_l  - q_l) - m_l)^2 \\
& \qquad \times \exp \left(  - \sum_l \tilde{\psi}_l (r_l, \tilde q_l , q_l) -4 r  \cdot (M) r\right) d\tilde q dq dr.
\end{align*}
Because the functions $\tilde{\psi}_l$ are symmetric in the variables $\tilde q_l$ and $q_l$ any term with an odd exponent vanishes. Hence, the exponents $n_1 ,\ldots, n_k, $ and $\tilde n_1 ,\ldots, \tilde n_k, $ are all even. Because $-M_{mn} \geq 0 $ due to the fact that the interaction is ferromagnetic we get
\begin{align*}
& -M_{mn}  \int  q_{l_1}^{n_1} \cdots q_{l_k}^{n_k} \tilde q_{\tilde l_1}^{\tilde n_1} \ldots \tilde q_{\tilde l_1}^{\tilde n_1}  ((r_l+\tilde q_l  - q_l) - m_l)^2 \\
& \qquad \times \exp \left(  - \sum_l \tilde{\psi}_l (r_l, \tilde q_l , q_l) -4 r  \cdot (M) r\right) d\tilde q dq dr \geq 0.
\end{align*}
All in all, the last inequality yields the desired estimate~\eqref{e_cond_covariances_crucial_estimate} and therefore completes the proof.
\end{proof}

\section{The Logarithmic Sobolev inequality: proof of Theorem~\ref{p_mr_OR}}\label{s_LSI}

This section is devoted to the proof of Theorem~\ref{p_mr_OR}.  We adapt the strategy of Otto \& Reznikoff~\cite[Theorem 3]{OR07} to our situation.
Recall that compared to Theorem~\ref{p_mr_OR}, we work with weaker assumptions:
\begin{itemize}
\item The single-site potentials $\psi_i$ are only quadratic and not super-quadratic (cf.~\eqref{e_cond_psi} vs.~\eqref{e_single_site_potential_otto}). Also note that in Theorem~\ref{p_OR_original} it is assumed that $M_{ii}=0$, whereas in Theorem~\ref{p_mr_OR} it is assumed that $M_{ii} \geq c >0$ (cf.~\eqref{e_strictly_diag_dominant}). In order to compare both statements it makes sense to think of the single-site potentials in Theorem~\ref{p_mr_OR} as
  \begin{align*}
    \psi_i (x_i) + \frac{1}{2} M_{ii} x_i^2.
  \end{align*}
 \item The interactions $M_{ij}$ decay only algebraically and not exponentially (cf.~\eqref{e_decay_inter_Otto} vs.~\eqref{e_cond_inter_alg_decay_OR}).
\item The correlations are decaying only algebraically and not exponentially (cf. \eqref{e_decay_corr_Otto}~vs.~\eqref{e_cond_alg_decay_OR}).
\end{itemize}
The algebraic decay of interactions and correlations is easy to incorporate in the original argument of~\cite{OR07}, whereas using quadratic and not super-quadratic potentials represents the main technical challenge of the proof. \medskip

The crucial ingredients in the proof of~\cite[Theorem 3]{OR07} are two auxiliary lemmas, namely \cite[Lemma 3 and Lemma 4]{OR07}. A careful analysis of the proof of~\cite{OR07} shows that only this part of the argument is sensitive to weakening the assumptions. Once the analog statements under weaker assumptions (see Lemma~\ref{p_crucial_lemma _1_OR} and Lemma~\ref{p_crucial_lemma _1_OR} below) are verified, the rest of the argument of~\cite[Theorem 3]{OR07} would work the same and is skipped in this article. The remaining part of the argument is based on an recursive application of a general principle, namely the two-scale criterion for LSI (cf.~\cite[Theorem 1]{OR07}), and is therefore not sensitive to changing the assumptions. Hence for the proof of Theorem~\ref{p_mr_OR} it suffices to show that the auxiliary lemmas \cite[Lemma 3 and Lemma 4]{OR07} remain valid under weakening the assumptions. \medskip

Let us turn to the first auxiliary Lemma (cf.~\cite[Lemma~3]{OR07} or Lemma~\ref{p_crucial_lemma _1_OR} from below). It states that the single-site conditional measures satisfy a LSI uniformly in the in the system size and the conditioned spin-values. The argument of \cite[Lemma~3]{OR07} by Otto \& Reznikoff is heavily based on the assumption that the single-site potential $\psi$ is super-quadratic. At this point we provide a new, different, and more elaborated argument showing that the statement of~\cite[Lemma~3]{OR07} remains valid if the single-site potential~$\psi$ is only perturbed quadratic. One could say that the proof of Lemma~\ref{p_crucial_lemma _1_OR} represents the main new ingredient compared to the argument of~\cite{OR07}.

\begin{lemma}[Generalization of~\mbox{\cite[Lemma~3]{OR07}}]\label{p_crucial_lemma _1_OR}

We assume the same conditions as in Theorem~\ref{p_mr_OR}. We consider for an arbitrary subset $S \subset \Lambda_{\mathrm{tot}}$ and site $i \in S$ the single-site conditional measure
  \begin{align*}
    \bar \mu (dx_i | x^S) := \frac{1}{Z} \exp (- \bar H ( (x^S)) d x_i
  \end{align*}
with Hamiltonian
\begin{align}\label{d_coarse_grained_hamiltonian}
  \bar H (  x^S) = - \log \int \exp(- H(x))  dx^{\Lambda_{\mathrm{tot}} \backslash S}.
\end{align}
Then the single-site conditional measure $\bar \mu (dx_i | x^S)$ satisfies a LSI with constant $\varrho>0$ (cf.~Definition~\ref{d_LSI}) that is uniform in $\Lambda_{\mathrm{tot}}$, $S$ and the conditioned spins~$x^S$.
\end{lemma}
We state the proof of Lemma~\ref{p_crucial_lemma _1_OR} in Section~\ref{s_crucial_lemma_1_OR}.  \medskip

Let us turn to the second auxiliary Lemma (cf.~\cite[Lemma~4]{OR07} or Lemma~\ref{p_crucial_lemma _2_OR} from below). For some fixed but large enough integer $K$ let us consider the $K$-sublattice $\Lambda_K$ given by
\begin{align}\label{e_def_sublattice}
  \Lambda_K := K \mathbb{Z}^d \cap \Lambda_{\mathrm{tot}}.
\end{align}
Let S an arbitrary subset satisfying $\Lambda_K \subset S \subset \Lambda_{\mathrm{tot}}$. The second auxiliary lemma states that measure on $\Lambda_K$, which is conditioned on the spins in $S \backslash \Lambda_{K}$ and averaged over the spins in $\Lambda_{\mathrm{tot}} \backslash S$, satisfies a LSI with constant $\varrho>0$ uniformly in $S$ and the conditioned spins:
\begin{lemma}[Generalization of~\mbox{\cite[Lemma~4]{OR07}}]\label{p_crucial_lemma _2_OR}
We assume the same conditions as in Theorem~\ref{p_mr_OR}. Let $S$ be an arbitrary set with $\Lambda_K \subset S \subset \Lambda_{\mathrm{tot}}$. Consider the conditional measure
  \begin{align*}
    \bar \mu (dx^{\Lambda_K} | x^{ S \backslash \Lambda_K}) := \frac{1}{Z} \exp (- \bar H ( x^S))  d x^ \Lambda_K
  \end{align*}
with Hamiltonian
\begin{align*
  \bar H (  x^S) = - \log \int \exp(- H(x))  dx^{\Lambda_{\mathrm{tot}} \backslash S}.
\end{align*}
Then there is some integer $K$ such that the conditional measure $ \bar \mu (dx^{\Lambda_K} | x^{S \backslash \Lambda_K})$ satisfies a LSI with constant $\varrho>0$ (cf.~Definition~\ref{d_LSI}) that is uniform in $\Lambda_{\mathrm{tot}}$, $S$ and the conditioned spins~$x^{S \backslash \Lambda_K}$.
\end{lemma}

\subsection{Proof of Lemma~\ref{p_crucial_lemma _1_OR} and Lemma~\ref{p_crucial_lemma _2_OR}}\label{s_crucial_lemma_1_OR}
Let us first turn to the proof of Lemma~\ref{p_crucial_lemma _1_OR}. For the argument we need the two new ingredients. The first one is the covariance estimate of Proposition~\ref{p_algebraic_decay_correlations} from below. The second one is that the variances of our kind of Gibbs measure are uniformly bounded (see Lemma~\ref{p_est_var_ss} from below). \medskip

Let us now state the covariance estimate of Proposition~\ref{p_algebraic_decay_correlations}.

\begin{proposition}\label{p_algebraic_decay_correlations}
Let $\Lambda \subset \mathds{Z}^d$ an arbitrary finite subset of the $d$-dimensional lattice $\mathds{Z}^d$. We consider a probability measure $d\mu:= Z^{-1} \exp (-H(x)) \ dx$ on $\mathds{R}^\Lambda$. We assume that
\begin{itemize}
\item the conditional measures $\mu(dx_i | \bar x_i )$, $i \in \Lambda $, satisfy a uniform PI with constant $\varrho_i>0$.
\item the numbers $\kappa_{ij}$, $i \neq j, i,j \in \Lambda$, satisfy
   \begin{equation*}
|\nabla_i \nabla_j H(x)|\leq \kappa_{ij}  < \infty
   \end{equation*}
uniformly in $x \in \mathds{R}^\Lambda$. Here, $|\cdot|$ denotes the operator norm of a bilinear form.
\item the numbers $\kappa_{ij}$ decay algebraically in the sense of
  \begin{align}
    \label{e_algeb_decay_of_kappa}
    \kappa_{ij} \lesssim \frac{1}{|i-j|^{d+\alpha} +1}
  \end{align}
for some $\alpha>0$.
 \item the symmetric matrix $A=(A_{ij})_{N \times N}$ defined by
  \begin{equation*}
A_{ij} =
\begin{cases}
  \varrho_i, & \mbox{if }\;  i=j , \\
  -\kappa_{ij}, & \mbox{if } \; i< j,
\end{cases}
  \end{equation*}
is strictly diagonally dominant  i.e.~for some $\delta > 0$ it holds for any $i \in \Lambda$
\begin{equation}\label{e_strictly_diag_dominant_A}
  \sum_{j \in \Lambda, j \neq i} |A_{ij}| + \delta \le A_{ii}.
\end{equation}
\end{itemize}
Then for all functions $f=f(x_i)$ and $g=g(x_j)$, $i, j \in \Lambda$,
   \begin{equation}
     \label{e_covariance_decay_algebraic}
      | \cov_{\mu}(f,g) | \lesssim (A^{-1})_{ij}   \left( \int |\nabla_i f|^2 \ d \mu \right)^{\frac{1}{2}} \left( \int |\nabla_j g |^2 \ d \mu \right)^{\frac{1}{2}}
   \end{equation}
and for any $i, j \in \Lambda$
\begin{align}
      \label{e_decay_M_inverse}
      |(A^{-1})_{ij}| \lesssim \frac{1}{|i-j|^{d + \tilde \alpha}+1},
    \end{align}
    for some $\tilde \alpha >0$.
  \end{proposition}
For the proof of Proposition~\ref{p_algebraic_decay_correlations} we refer the reader to the article~\cite{Cov_est}.

\begin{proof}[Proof of Lemma~\ref{p_crucial_lemma _1_OR}]

The strategy is to show that the Hamiltonian~$\bar H_i (  x_i )$ of the single-site conditional measure~$  \bar \mu (dx_i | x^S)$ is perturbed strictly- convex in the sense that there exists a splitting
\begin{align}
  \label{e_decom_single_site_hamitlonian}
  \bar H_i (  x_i) = \tilde \psi_i^c (x_i) + \tilde \psi_i^b (x_i)
\end{align}
into the sum of two functions $\tilde \psi_i^c (x_i)$ and $\tilde \psi_i^b (x_i)$ satisfying
\begin{align}
  \label{e_ssp_perturbed_strictly_convex}
  \tilde (\psi_i^c)'' (x_i) \geq c >0 \quad \mbox{and} \quad |\tilde \psi_i^b (x_i)| \leq C < \infty
\end{align}
uniformly in $x_i \in \mathbb{R}$, $i\in S$, $\Lambda_{\mathrm{tot}}$ and $S$.\newline
Once~\eqref{e_decom_single_site_hamitlonian} and~\eqref{e_ssp_perturbed_strictly_convex} are validated, the statement of Lemma~\ref{p_crucial_lemma _1_OR} follows simply from a combination of the criterion of Bakry-\'Emery for LSI and the Holley-Stroock perturbation principle (cf.~Appendix~\ref{s_BE_HS} and the proof of~\cite[Lemma 1]{OR07} for details).  \medskip

The aim is to decompose $\bar H_i$ such that~\eqref{e_decom_single_site_hamitlonian} and~\eqref{e_ssp_perturbed_strictly_convex} is satisfied. For that purpose, let us define the auxiliary Hamiltonian $H_{\mathrm{aux}} (x)$, $x \in \mathbb{R}^{\Lambda_{\mathrm{tot}}}$, as
\begin{align}\label{e_def_H_aux}
  H_{\mathrm{aux}} (x) = H(x) - \sum_{j: |j-i| \leq R} \psi_i^b (x_j).
\end{align}
Note that $H_{\mathrm{aux}}$ is strictly convex, if restricted to spins $x_j$ with $|i-j| \leq R$.\newline
For convenience, let us introduce the notation $S^c := \Lambda_{\mathrm{tot}} \backslash S$. The Hamiltonian $\bar H_i$ is then written as
\begin{align*}
  \bar H_i (x_{i}) & \overset{\eqref{d_coarse_grained_hamiltonian}}{=} - \log \int \exp(- H(x)) dx^{S^c} \\
  & = \underbrace{- \log \int \exp(- H_{\mathrm{aux}}(x)) dx^{S^c}}_{=: \tilde \psi_i^c (x_i)}\\
  & \qquad \underbrace{- \log \frac{\int \exp(- H(x)) dx^{S^c}}{\int \exp(- H_{\mathrm{aux}}(x)) dx^{S^c}}}_{=: \tilde \psi_i^b (x_i) }.
\end{align*}
Now, let us check that the functions $\tilde \psi_i^c (x_i)$ and $\tilde \psi_i^b (x_i)$ defined by the last identity satisfy the structural condition~\eqref{e_ssp_perturbed_strictly_convex}. \medskip

Let us consider first the function $\tilde \psi_i^b (x_i)$.  We introduce the auxiliary measure $\mu_{\mathrm{aux}}$ by
\begin{align*}
  \mu_{\mathrm{aux}} (dx^{S^c}) = \frac{1}{Z} \exp \left( - H_{\mathrm{aux}} (x) \right) dx^{S^c}.
\end{align*}
Then it follows from the definition~\eqref{e_def_H_aux} of $H_{\mathrm{aux}}$ that
\begin{align*}
  \left| \tilde \psi_i^b (x_i) \right| & \leq \left| \log \int \exp (- \sum_{j:|j-i| \leq R} \psi_i^b (x_j)) \ \mu_{\mathrm{aux}} (dx^{S^c}) \right| \\
  & \leq \sum_{j:|j-i|\leq R} \| \psi_i^b \|_{\infty} \leq 2(R+1)^d C.
\end{align*}

It is now left to show that~$\tilde \psi_i^c (x_i)$ is uniformly strictly convex. Direct calculation yields
\begin{align}
  \label{e_second_deriv_tilde_psi_c}
  \frac{d^2}{dx_i^2} \tilde \psi_i^c (x_i) &= \int \frac{d^2}{dx_i^2} H_{\mathrm{aux}} (x) \mu_{\mathrm{aux}} - \var_{\mu_{\mathrm{aux}}} \left( \frac{d}{dx_i} H_{\mathrm{aux}} (x) \right) .
\end{align}
We decompose the measure $\mu_{\mathrm{aux}}$ into
\begin{align*}
&  \mu_{\mathrm{aux}} ( dx^{S^c}) \\
 & \quad = \mu_{\mathrm{aux}} \left( (dx_j)_{j \in S^c, |j-i| \leq R} \ | \ (x_j)_{j \in S^c, |j-i|> R}  \right ) \bar \mu_{\mathrm{aux}} ( (dx_j)_{j \in S^c, |j-i|> R}).
\end{align*}
Here, $\mu_{\mathrm{aux}} \left( (dx_j)_{j \in S^c, |j-i| \leq R} \ | \ (x_j)_{j \in S^c, |j-i | > R}  \right )$ denotes the conditional measure given by
\begin{align*}
 & \mu_{\mathrm{aux}} \left(  (dx_j)_{j \in S^c, |j-i| \leq R} \ | \ (x_j)_{j \in S^c, |j-i|> R}  \right ) \\
 & \qquad = \frac{1}{Z} \exp\left( - H_{\mathrm{aux}} (x) \right) \   \otimes_{\substack{j \in S^c, \\ |j-i| \leq R}}  dx_j ,
\end{align*}
whereas $\bar \mu_{\mathrm{aux}} ( (dx_j)_{j \in S^c, |j-i|> R})$ denotes the marginal measure given by
\begin{align*}
&  \bar \mu_{\mathrm{aux}} ( (dx_j)_{j \in S^c, |j-i|> R }) \\ & \qquad = \frac{1}{Z} \left( \int \exp\left( - H_{\mathrm{aux}} (x) \right) \   \otimes_{\substack{l \in  S^c, \\ |l-i| \leq R}}  dx_l \right) \ \ \otimes_{\substack{j \in S^c, |j-i|> R }}  dx_j  .
\end{align*}
For convenience, we write $\mu_{\mathrm{aux},c}$ instead of the conditional measure $\mu_{\mathrm{aux}} \left(  (dx_j)_{j \in S^c, |j-i| \leq R} \ | \ (x_j)_{j \in S^c, |j-i|> R}  \right )$.
Applying the decomposition to~\eqref{e_second_deriv_tilde_psi_c} yields
\begin{align}
  \frac{d^2}{dx_i^2} \tilde \psi_i^c (x_i) &= \int \left(  \int  \frac{d^2}{dx_i^2} H_{\mathrm{aux}} (x) \mu_{\mathrm{aux},c}   - \var_{\mu_{\mathrm{aux},c}} \left( \frac{d}{dx_i} H_{\mathrm{aux}} (x) \right)  \right)  \  \bar \mu_{\mathrm{aux}} \notag \\
& \qquad  - \var_{\bar \mu_{\mathrm{aux}}} \left( \int \frac{d}{dx_i} H_{\mathrm{aux}} (x) \mu_{\mathrm{aux},c}\right). \label{e_decomp_desintegration}
\end{align}
The first term on the right hand side of the last identity is controlled easily. Note that the Hamiltonian $H_{\mathrm{aux}}$ is strictly-convex, if restricted to spins $x_j$ with $|j-i|\leq R$. So it follows from a standard argument based on the Brascamp lieb inequality that (for details see for example~\cite[Chapter 3]{Dizdar})
\begin{align*}
   \int  \frac{d^2}{dx_i^2} H_{\mathrm{aux}} (x) \mu_{\mathrm{aux},c}   - \var_{\mu_{\mathrm{aux},c}} \left( \frac{d}{dx_i} H_{\mathrm{aux}} (x) \right)  \geq c >0
\end{align*}
uniformly in $R$ and therefore also
\begin{align*}
  \int \left(  \int  \frac{d^2}{dx_i^2} H_{\mathrm{aux}} (x) \mu_{\mathrm{aux},c}   - \var_{\mu_{\mathrm{aux},c}} \left( \frac{d}{dx_i} H_{\mathrm{aux}} (x) \right)  \right)  \  \bar \mu_{\mathrm{aux}} \geq c >0
\end{align*}
uniformly in $R$. \medskip

Let us now turn to the second term in~\eqref{e_decomp_desintegration}. Straightforward calculation yields
\begin{align*}
  \frac{d}{dx_i} H_{\mathrm{aux}} (x) & = (\psi_i^c)' (x_i) + M_{ii} x_i + s_i + \frac{1}{2}  \sum_{j \in \Lambda_{\mathrm{aux}}} M_{ij} x_j.
\end{align*}
Because the measures $\bar \mu_{\mathrm{aux}}$ and $\mu_{\mathrm{aux},c}$ live on a subset of $S^c$, $i \in S$, and the variance is invariant under adding constants, we have
\begin{align}
&  \var_{\bar \mu_{\mathrm{aux}}} \left( \int \frac{d}{dx_i} H_{\mathrm{aux}} (x) \mu_{\mathrm{aux},c}\right) = \var_{\bar \mu_{\mathrm{aux}}} \left( \frac{1}{2} \int \sum_{j \in S^c} M_{ij} x_j \  \mu_{\mathrm{aux},c}\right) \notag \\
  & \quad = \frac{1}{4}\var_{\bar \mu_{\mathrm{aux}}} \left( \sum_{\substack{j \in S^c, \\ |j-i| > R}} M_{ij} x_j \right)  \\
& \qquad + \frac{1}{4}\var_{\bar \mu_{\mathrm{aux}}} \left( \int \sum_{\substack{j \in S^c , \\ |j-i| \leq R}} M_{ij} x_j \  \mu_{\mathrm{aux},c}\right). \label{e_decomp_var_crucail_lemma_1_OR}
\end{align}
The first summand on the right hand side of the last identity is estimated in a straightforward manner i.e.
\begin{align*}
&  \var_{\bar \mu_{\mathrm{aux}}} \left( \sum_{\substack{j \in S^c, \\ |j-i| > R}} M_{ij} x_j \right) \\
& = \var_{ \mu_{\mathrm{aux}}} \left( \sum_{\substack{j \in S^c, \\ |j-i| > R}} M_{ij} x_j \right) \\
 & =  \sum_{\substack{j \in S^c, \\ |j-i| > R}} M_{ij} \sum_{\substack{l \in S^c, \\ |l-i| > R}} M_{il} \cov_{\mu_{\mathrm{aux}}} (x_j,x_l) \\
& \leq \sum_{\substack{j \in S^c, \\ |j-i| > R}} \sum_{\substack{l \in S^c, \\ |l-i| > R}} M_{ij}  M_{il} \left( \var_{\mu_{\mathrm{aux}}} (x_j) \right)^{\frac{1}{2}}  \left( \var_{\mu_{\mathrm{aux}}} (x_l) \right)^{\frac{1}{2}}\\
& \overset{\eqref{e_est_ss_var}}{\leq}  C \sum_{\substack{j \in S^c, \\ |j-i| > R}} \sum_{\substack{l \in S^c, \\ |l-i| > R}} M_{ij}  M_{il} \\
& \overset{~\eqref{e_cond_inter_alg_decay_OR}}{\leq}  C \sum_{\substack{j \in S^c, \\ |j-i| > R}} \sum_{\substack{l \in S^c, \\ |l-i| > R}} \frac{1}{|i-j|^{d + \alpha} +1 } \ \frac{1}{|i-l|^{d + \alpha} +1 }   \\
& \leq C \frac{1}{R^{\frac{\alpha}{2}}}.
\end{align*}
Here we have used one of the new ingredients, namely the uniform estimate~\eqref{e_est_ss_var} stated in Lemma~\ref{p_est_var_ss} from below. Note that Lemma~\ref{p_est_var_ss} also applies to the measure~$\mu_{\mathrm{aux}}$ because~$\mu_{\mathrm{aux}}$ satisfies the same structural assumptions as the measure~$\mu_{\Lambda}$. \newline
Let us consider now the second summand on the right hand side of~\eqref{e_decomp_var_crucail_lemma_1_OR}. By doubling the variables we get
\begin{align*}
 & \var_{\bar \mu_{\mathrm{aux}}} \left( \int \sum_{\substack{j \in S^c , \\ |j-i| \leq R}} M_{ij} x_j \  \mu_{\mathrm{aux},c}\right) \\
 & \quad = \int \Big( \int \sum_{\substack{j \in S^c , \\ |j-i| \leq R}} M_{ij} x_j \  \mu_{\mathrm{aux},c} (dx| y) \\
& \quad \qquad - \int \sum_{\substack{j \in S^c , \\ |j-i| \leq R}} M_{ij} x_j \  \mu_{\mathrm{aux},c}  (dx| \bar y )\Big)^2 \bar \mu_{\mathrm{aux}} (dy)  \bar \mu_{\mathrm{aux}} (d \bar y)
\end{align*}
By interpolation we have
\begin{align*}
&   \int \sum_{\substack{j \in S^c , \\ |j-i| \leq R}} M_{ij} x_j \  \mu_{\mathrm{aux},c} (dx| y)- \int \sum_{\substack{j \in S^c , \\ |j-i| \leq R}} M_{ij} x_j \  \mu_{\mathrm{aux},c}  (dx| \bar y )  \\
& \quad = \int_0^1 \frac{d}{dt} \int \sum_{\substack{j \in S^c , \\ |j-i| \leq R}} M_{ij} x_j \  \mu_{\mathrm{aux},c} (dx| ty + (1-t) \bar y) \ dt \\
& \quad = \int_0^1 \cov_{\mu_{\mathrm{aux},c} (dx| ty + (1-t) \bar y)} \left( \sum_{\substack{j \in S^c , \\ |j-i| \leq R}} M_{ij} x_j, \sum_{\substack{k,l \in S^c , \\ |k-i| \leq R \\ |l-i| \geq R }} x_k M_{kl} (\bar y_l - y_l) \right) \ dt \\
& \quad = \int_0^1 \sum_{\substack{j \in S^c , \\ |j-i| \leq R}} M_{ij} M_{kl} (\bar y_l - y_l) \sum_{\substack{k,l \in S^c , \\ |k-i| \leq R \\ |l-i| \geq R }}  \cov_{\mu_{\mathrm{aux},c} (dx| ty + (1-t) \bar y)} \left(  x_j,  x_k  \right) \ dt.
\end{align*}
Without loss of generality we may assume that the interaction is ferromagnetic i.e.~$M_{kl} \leq 0$ for all $k \neq l$ (else use $M_{kl}\leq|M_{kl}|$ and Lemma~\ref{p:attractive_interact_dominates}). Note that the measure $\mu_{\mathrm{aux},c}$ has strictly convex single-site potentials. Therefore the single-site conditional measures $\mu (dx_1| x)$ satisfy a LSI with constant $\frac{1}{2}M_{ii}$ by the Bakry-\'Emery criterion (see Theorem~\ref{local:thm:BakryEmery}). Because the interaction is strictly-diagonally dominant in the sense of~\eqref{e_strictly_diag_dominant}, an application of Proposition~\ref{p_algebraic_decay_correlations} yields that the covariance can be estimated as
  \begin{align*}
    \cov_{\mu_{\mathrm{aux},c} (dx| ty + (1-t) \bar y)} \left(  x_j,  x_k  \right) \leq (M^{-1})_{jk},
  \end{align*}
where the matrix $M$ is given by the elements
\begin{align*}
  M_{ln} \quad \mbox{for} \quad  l , n \in S^c , |l-i| \leq R, |k-i| \leq R  \quad \mbox{or} \quad l=n=i.
\end{align*}
We want to note that by an simple standard result (see for example~\cite[Lemma 5]{OR07}) or~\cite[Lemma 4.3]{MN}) it holds $(M^{-1})_{kl} \geq 0$ for all $k,l$. Using this information, we get by an application of Jensen's inequality that
  \begin{align*}
 & \var_{\bar \mu_{\mathrm{aux}}} \left( \int \sum_{\substack{j \in S^c , \\ |j-i| \leq R}} M_{ij} x_j \  \mu_{\mathrm{aux},c}\right) \\
& \quad \leq C  \int_0^1 \sum_{\substack{j \in S^c , \\ |j-i| \leq R}}  \sum_{\substack{k,l \in S^c , \\ |k-i| \leq R \\ |l-i| \geq R }}  M_{ij} M_{kl}   (M^{-1})_{jk} \int (\bar y_l - y_l)^2 \ \bar \mu_{\mathrm{aux}} (dy) \bar \mu_{\mathrm{aux}} (d \bar y)\ dt \ \\
& \quad \leq C   \sum_{\substack{j \in S^c , \\ |j-i| \leq R}}  \sum_{\substack{k,l \in S^c , \\ |k-i| \leq R \\ |l-i| \geq R }}  M_{ij} M_{kl}   (M^{-1})_{jk} \var_{\bar \mu_{\mathrm{aux}}}( y_l)\\
& \quad \leq C  \sum_{\substack{j \in S^c , \\ |j-i| \leq R}} \sum_{\substack{k,l \in S^c , \\ |k-i| \leq R \\ |l-i| \geq R }}  M_{ij} M_{kl}  (M^{-1})_{jk} \var_{ \mu_{\mathrm{aux}}}( y_l)\\
& \quad \overset{\eqref{e_est_ss_var}}{\leq} C   \sum_{\substack{j \in S^c , \\ |j-i| \leq R}}  \sum_{\substack{k,l \in S^c , \\ |k-i| \leq R \\ |l-i| \geq R }}  M_{ij} M_{kl}  (M^{-1})_{jk} \\
& \quad \overset{~\eqref{e_cond_inter_alg_decay_OR}, \eqref{e_decay_M_inverse}}{\leq} C   \sum_{\substack{j \in S^c , \\ |j-i| \leq R}}  \sum_{\substack{k,l \in S^c , \\ |k-i| \leq R \\ |l-i| \geq R }} \frac{1}{|i-j|^{d+\alpha} +1} \ \frac{1}{|k-l|^{d+\alpha} +1} \frac{1}{|j-k|^{d+\tilde \alpha} +1} \\
& \quad \leq \frac{C}{R^{\frac{\tilde \alpha}{{2}}}}
\end{align*}
Note that here we also used the second ingredient, namely the covariance estimates~\eqref{e_covariance_decay_algebraic} and~\eqref{e_decay_M_inverse}.
Hence, both terms on the right hand side of~\eqref{e_decomp_var_crucail_lemma_1_OR} are arbitrarily small, if we choose $R$ big enough. Overall this leads to the desired statement (cf.~\eqref{e_decomp_desintegration} ff.)
\begin{align*}
  \frac{d^2}{dx_i^2} \tilde \psi_i^c (x_i) \geq c >0,
\end{align*}
which completes the argument.
\end{proof}

In the proof of Lemma~\ref{p_crucial_lemma _1_OR}, we needed the following auxiliary statement.
\begin{lemma}\label{p_est_var_ss}
  Under the same assumptions as in Lemma~\ref{p_crucial_lemma _1_OR}, it holds that for all $i \in \Lambda$
  \begin{align}\label{e_est_ss_var}
    \var_{\mu_\Lambda} (x_i) \leq C,
  \end{align}
where the bound is uniform in $\Lambda$ and only depends on the constants appearing in~\eqref{e_cond_psi} and in~\eqref{e_strictly_diag_dominant}.
\end{lemma}
The proof of Lemma~\ref{p_est_var_ss} is a simple and straightforward application of a exponential moment bound due to Robin Nittka.
\begin{lemma}[\mbox{\cite[Lemma~4.3]{MN}}]\label{lem:moments}
  We assume that the formal Hamiltonian $H:\mathds{R}^{\mathds{Z}^d} \to \mathds{R} $ given by~\eqref{e_d_Hamiltonian} satisfies the Assumptions~\eqref{e_cond_psi}~-~\eqref{e_strictly_diag_dominant}.\newline
Additionally, we assume that for all $ i \in \mathds{Z}^d$ the convex part $\psi_i^c$ of the single-site potentials $\psi_i$ has a global minimum in $x_i=0$. \newline
Let $\delta >0$ be given by~\eqref{e_strictly_diag_dominant}. Then for every $0 \le a \le \frac{\delta}{2}$ and any subset $\Lambda \subset \mathds{Z}^d$ it holds
        \begin{equation} \label{e:exponential_moment}
          \mathds{E}_{\mu_{\Lambda}} \bigl[\mathrm{e}^{a p_i^2}\bigr] \lesssim 1.
        \end{equation}
In particular, for any $k \in \mathds{N}_0$ this yields
\begin{equation} \label{e:arbitrary_moment}
  \mathds{E}_{\mu_{\Lambda}}[p_i^{2k}] \lesssim k!.
\end{equation}
\end{lemma}
The statement of Lemma~\ref{lem:moments} is a slight improvement of~\cite[Section~3]{BHK82}, because the assumptions are slightly weaker compared to~\cite {BHK82}. More precisely, $\psi_i''$ may change sign outside every compact set and there is no condition on the signs of the interaction. Even if \cite[Lemma~4.3]{MN} is formulated in~\cite{MN} for systems on an one-dimensional lattice, a simple analysis of the proof shows that the statement is also true on lattices of any dimension.
\begin{proof}[Proof of Lemma~\ref{p_est_var_ss}]
By doubling the variables we get
\begin{equation*}
  \var_{\mu_{\Lambda}}(x_i) = \frac{1}{2} \int \int (x_i -y_i)^2  \mu_{\Lambda}(dx) \mu_{\Lambda} (dy).
\end{equation*}
By the change of coordinates $x_k= q_k + p_k$ and $y_k= q_k -p_k$ for all $k \in \Lambda$, the last identity yields by using the definition~\eqref{e_d_Gibbs_measure} of the finite-volume Gibbs measure $\mu_{\Lambda}$ that
\begin{align*}
  \var_{\mu_{\Lambda}}(x_i) &= C \int \int p_i^2  \  \underbrace{\frac{e^{-H(q^{\Lambda}+p^{\Lambda}, x^{\mathds{Z}\backslash \Lambda}) - H(q^{\Lambda}-p^{\Lambda}, x^{\mathds{Z}\backslash \Lambda})}}{\int e^{-H(q^{\Lambda}+p^{\Lambda}, x^{\mathds{Z}\backslash \Lambda})  -H(q^{\Lambda}-p^{\Lambda}, x^{\mathds{Z}\backslash \Lambda})}  \mathrm{d} p^{\Lambda} \mathrm{d} q^{\Lambda} }   \mathrm{d} p^{\Lambda} \mathrm{d} q^{\Lambda} }_{=: d \tilde \mu_{\Lambda} (q^{\Lambda},p^{\Lambda})} .
\end{align*}
By conditioning on the values $q^{\Lambda}$ it directly follows from the definition~\eqref{e_d_Hamiltonian} of $H$ that
\begin{equation} \label{e:repre_covariance}
  \var_{\mu_{\Lambda}}(x_i) = C
 \mathds{E}_{\tilde \mu_{\Lambda}} \left[ \mathds{E}_{\mu_{\Lambda,q}} \left[ p_i^2 \right] \right].
\end{equation}
Here, the conditional measure $\mu_{\Lambda,q}$ is given by the density
\begin{equation}
  \label{eq:def_mu_q}
	\mathrm{d}\mu_{\Lambda,q}(p^\Lambda) \coloneqq \frac{1}{Z_{\mu_{\Lambda,q}}} \mathrm{e}^{-\sum_{k \in \Lambda} \psi_{k,q}(p_k) - \sum_{k,l \in \Lambda} M_{kl} p_k p_l} \mathrm{d} p^\Lambda
\end{equation}
with single-site potentials $\psi_{k,q} \coloneqq \psi_{k,q}^c + \psi_{k,q}^b$ defined by
\begin{align*}
   \psi_{k,q}^c(p_k) & \coloneqq \psi_k^c(q_k + p_k) + \psi_k^c(q_k - p_k) \qquad \mbox{and} \\
  \psi_{k,q}^b(p_k) & \coloneqq \psi_k^b(q_k + p_k) + \psi_k^b(q_k - p_k).
\end{align*}
Because of symmetry in the variable $p_k$,the convex part of the single-site potential $\psi_{k,q}^c(p_k)$ has a global minimum at $p_k=0$ for any $k$. Therefore, an application of Lemma~\ref{lem:moments} yields the desired statement. \end{proof}

Let us turn to the verification of Lemma~\ref{p_crucial_lemma _2_OR}. We also need an auxiliary statement, namely Lemma~\ref{p_aux_lemma_2_OR} from below.
It is a generalization of~\cite[Lemma 2]{OR07} and states that the interactions of the Hamiltonian $\bar H ((x_i)_{i \in S})$ given by~\eqref{d_coarse_grained_hamiltonian} decay sufficiently fast.
\begin{lemma} [Generalization of~\mbox{\cite[Lemma 2]{OR07}}]\label{p_aux_lemma_2_OR}
In the same situation as in Lemma~\ref{p_crucial_lemma _1_OR}, the interactions of $\bar H (x^S)$ decay algebraically i.e. there are constants $0, \varepsilon, C < \infty$  such that
\begin{align*}
  \left|\frac{d}{dx_i} \frac{d}{dx_j}  \bar H \right| \leq C \frac{1}{|i-j|^{d+\bar \varepsilon}+1}
\end{align*}
uniformly in $i,j \in S$.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{p_aux_lemma_2_OR}]
  Direct calculation as in~\cite[Lemma 2]{OR07} shows that
\begin{align*}
  \frac{d}{dx_i} \frac{d}{dx_j}  \bar H = -M_{ij} - \sum_{k \in \Lambda_{\mathrm{tot}} \backslash S}  \sum_{l \in \Lambda_{\mathrm{tot}} \backslash S} M_{ik} \ M_{jl} \  \cov_{\Lambda_{\mathrm{tot} \backslash S}} (x_k, x_l).
\end{align*}
The last identity immediately yields the estimate (cf.~\cite[(52)]{OR07})
\begin{align*}
  \left|\frac{d}{dx_i} \frac{d}{dx_j}  \bar H \right| \leq |M_{ij}| + \sum_{k \in \Lambda_{\mathrm{tot}} \backslash S}  \sum_{l \in \Lambda_{\mathrm{tot}} \backslash S} |M_{ik}| \ |M_{jl}| \ | \cov_{\Lambda_{\mathrm{tot} \backslash S}} (x_k, x_l)|
\end{align*}
 Using the decay of interactions~\eqref{e_decay_inter_Otto} and the decay of correlations~\eqref{e_decay_corr_Otto} we get
\begin{align*}
 & \left|\frac{d}{dx_i} \frac{d}{dx_j}  \bar H \right| \\
 & \quad \leq \frac{C}{|i-j|^{d+ \alpha}} + C \sum_{k \in \Lambda_{\mathrm{tot}} \backslash S}  \sum_{l \in \Lambda_{\mathrm{tot}} \backslash S} \frac{1}{|i-k|^{d+ \alpha}}    \frac{1}{|j-l|^{d+ \alpha}}          \frac{1}{|k-l|^{d+ \alpha}}
\end{align*}
Now we use the same kind of argument as used in in the proof of Proposition~\ref{p_algebraic_decay_correlations} to estimate the term $T_k$. This means that for any multi-indexes $i,k,l,j \in \Lambda_{\mathrm{tot}}$ it holds either
\begin{align*}
  |i-k|\geq \frac{C}{3} |i-j|, \quad  |j-l|\geq \frac{C}{3} |i-j|, \quad \mbox{or} \quad |k-l|\geq \frac{C}{3} |i-j|.
\end{align*}
Therefore,  we have
\begin{align*}
& \sum_{k \in \Lambda_{\mathrm{tot}} \backslash S}  \sum_{l \in \Lambda_{\mathrm{tot}} \backslash S} \frac{1}{|i-k|^{d+ \alpha}}    \frac{1}{|j-l|^{d+ \alpha}}          \frac{1}{|k-l|^{d+ \alpha}}   \\
& \leq \sum_{ \substack{k \in \Lambda_{\mathrm{tot}} \backslash S, \\ l \in \Lambda_{\mathrm{tot}} \backslash S, \\ |i-k|\geq \frac{1}{3} |i-j|}} \frac{1}{|i-k|^{d + \alpha}}    \frac{1}{|j-l|^{d + \alpha}}          \frac{1}{|k-l|^{d + \alpha}} \\
& \qquad + \sum_{ \substack{k \in \Lambda_{\mathrm{tot}} \backslash S, \\ l \in \Lambda_{\mathrm{tot}} \backslash S, \\ |j-l|\geq \frac{1}{3} |i-j|}} \ldots + \sum_{ \substack{k \in \Lambda_{\mathrm{tot}} \backslash S, \\ l \in \Lambda_{\mathrm{tot}} \backslash S, \\ |k-l|\geq \frac{1}{3} |i-j|}} \ldots \\
& \leq C \ \frac{1}{|i-j|^{d + \alpha}},
\end{align*}
which yields the desired statement of Lemma~\ref{p_aux_lemma_2_OR}.
\end{proof}
As in the proof of~\cite[Lemma~4]{OR07} we verify Lemma~\ref{p_crucial_lemma _2_OR} by an application of the Otto-Reznikoff criterion for LSI i.e.
\begin{theorem}[Otto-Reznikoff criterion for LSI,~{\mbox{\cite[Theorem 1]{OR07}}}] \label{p_otto_reznikoff}
  Let $d\mu:= Z^{-1} \exp (-H(x)) \ dx$ be a probability measure on a direct product of Euclidean spaces $X= X_1 \times \cdots \times X_N$. We assume that
\begin{itemize}
 \item the conditional measures $\mu(dx_i | \bar x_i )$, $1\leq i \leq N$, satisfy a uniform LSI($\varrho_i $).
 \item the numbers $\kappa_{ij}$, $1 \leq i \neq j \leq N$, satisfy
   \begin{equation*}
 |\nabla_i \nabla_j H(x)|\leq \kappa_{ij} < \infty
   \end{equation*}
uniformly in $x \in X$. Here, $|\cdot|$ denotes the operator norm of a bilinear form.
\item the symmetric matrix $A=(A_{ij})_{N \times N}$ defined by
  \begin{equation*}
A_{ij} =
\begin{cases}
  \varrho_i, & \mbox{if } \; i=j , \\
  -\kappa_{ij}, & \mbox{if } \; i < j,
\end{cases}
  \end{equation*}
satisfies in the sense of quadratic forms
\begin{equation}\label{e_cond_OR}
   A \geq \varrho \Id \qquad \mbox{for a constant } \varrho>0.
\end{equation}
\end{itemize}
Then $\mu$ satisfies LSI($\varrho$).
\end{theorem}
\begin{proof}[Proof of Lemma~\ref{p_crucial_lemma _2_OR}]
  We want to apply Theorem~\ref{p_otto_reznikoff}. By an application of Lemma~\ref{p_crucial_lemma _1_OR}, we know that the single-site measures conditional measures $\bar \mu (dx_i |x^{\Lambda_K} x^{ S \backslash,  \Lambda_K})$, $i \in \Lambda_k$ satisfy a LSI with uniform constant $\varrho>0$. \newline
For the mixed derivatives of the Hamiltonian, we have according to Lemma~\ref{p_aux_lemma_2_OR}
\begin{align*}
  \left|\frac{d}{dx_i} \frac{d}{dx_j}  \bar H \right| \leq C \frac{1}{|i-j|^{d+\bar \varepsilon}+1}.
\end{align*}
Hence, in order to apply Theorem 1 we have to consider the symmetric matrix $A = (A_{ij})_{i,j \in \Lambda_K}$ with
  \begin{align*}
      & A_{ii}= \varrho, \\
      & A_{ij} = - C \frac{1}{|i-j|^{d+\bar \varepsilon} +1}, \qquad \mbox{for } i\neq j.
  \end{align*}
We will argue that A is strict positive-definite if we choose the integer $K$ large enough.
We have
\begin{align*}
  \sum_{i,j \in \Lambda_K} x_i A_{ij} x_j = \sum_{i \in \Lambda_K} \varrho x_i^2 + \sum_{i,j \in \Lambda_K, \ i \neq j} x_i A_{ij} x_j.
\end{align*}
Let us estimate the second term of the right hand side. We have
\begin{align*}
 | \sum_{i,j \in \Lambda_K, \ i \neq j} x_i A_{ij} x_j | & \leq \frac{1}{2} \sum_{i \in \Lambda_K} \sum_{j \in \Lambda_K, \ i \neq j} |A_{ij}| x_i^2 + \frac{1}{2} \sum_{j \in \Lambda_K} \sum_{i \in \Lambda_K, \ i \neq j} |A_{ij}| x_j^2\\
& \leq C \sum_{i \in \Lambda_K} \sum_{j \in \Lambda_K, \ i \neq j} \frac{1}{|i-j|^{d + \bar \varepsilon}} x_i^2 \\
 & \leq \frac{C}{K^{\frac{\bar \varepsilon}{2}}} \sum_{i \in \Lambda_K} x_i^2 \sum_{j \in \Lambda_K, \ i \neq j} \frac{1}{|i-j|^{d + \frac{\bar \varepsilon}{2}}}
\end{align*}
where the last inequality holds if we choose $K$ large enough. So we get overall that
\begin{align*}
  \sum_{i,j \in \Lambda_K} x_i A_{ij} x_j \geq \frac{\varrho}{2} \sum_{i \in \Lambda_K} x_i^2 >0 ,
\end{align*}
 which yields the desired statement of Lemma~\ref{p_crucial_lemma _2_OR} by an application of Theorem~\ref{p_otto_reznikoff}.

\end{proof}


\section{Introduction}
\label{sec:intro}
Nucleon polarizabilities are of fundamental importance for understanding the dynamics of the internal structure of nucleons. The static electric and magnetic dipole polarizabilities, $\alpha_{E1}$ and $\beta_{M1}$, characterize the response of the nucleon to external electromagnetic stimulus by relating the strength of the induced electric and magnetic dipole moments of the nucleon to the applied field. Decades-long endeavors have been devoted to studying the static polarizabilities of nucleons experimentally and theoretically~\cite{Schumacher05,Griesshammer12,Hagelstein15}. Nuclear Compton scattering is a powerful tool to access the nucleon polarizabilities, where incident real photons apply an electromagnetic field to the nucleon and induce multipole radiation by displacing the charges and currents inside the nucleon. In the past decade, effective field theories (EFTs) have proven to be successful theoretical frameworks to describe such processes as well as to predict and extract static nucleon polarizabilities from the low-energy Compton scattering data~\cite{Griesshammer12}. Values of $\alpha_{E1}$ and $\beta_{M1}$ of the proton have been successfully extracted from Compton scattering experiments using liquid hydrogen targets. With $\alpha_{E1}^p+\beta_{M1}^p$ constrained by the Baldin sum rule (BSR), the latest EFT fit to the global database of proton Compton scattering gives~\cite{McGovern12,Griesshammer15}
\begin{equation}
\begin{split}
\alpha_{E1}^p &= 10.65\pm0.35_{\rm stat}\pm0.2_{\rm BSR}\pm0.3_{\rm theo},\\
\beta_{M1}^p &= 3.15\mp0.35_{\rm stat}\pm0.2_{\rm BSR}\mp0.3_{\rm theo},
\end{split}
\end{equation}
where the polarizabilities are given here and throughout this paper in units of $10^{-4}\,\rm fm^3$. The neutron polarizabilities, in contrast, are less well determined due to the lack of free neutron targets and the small Thomson cross sections of the neutron due to the fact that the neutron is uncharged~\cite{Schumacher05,Myers12,Myers:2014ace}.

Light nuclear targets, such as liquid deuterium~\cite{Hornidge99,Lundin02,Myers15,Myers:2014ace}, liquid $^4$He~\cite{Sikora17,Fuhrberg95,Proff99}, and $^6$Li~\cite{Myers12,Myers14}, can be utilized as effective neutron targets to extract neutron polarizabilities. After accounting for the binding effects, these isoscalar targets allow for the extraction of the isoscalar-averaged polarizabilities of the proton and neutron. By subtracting the better-known proton results, the neutron polarizabilities can be obtained. Indeed, the group at the MAX~IV Laboratory in Lund reported the most recent EFT extraction of the neutron polarizabilities from the world data of elastic deuteron Compton scattering as~\cite{Myers:2014ace}
\begin{equation}
\begin{split}
\alpha_{E1}^n &= 11.55\pm1.25_{\rm stat}\pm0.2_{\rm BSR}\pm0.8_{\rm theo}, \\
\beta_{M1}^n &= 3.65\mp1.25_{\rm stat}\pm0.2_{\rm BSR}\mp0.8_{\rm theo},
\end{split}
\end{equation}
with the BSR constraint applied.

Although no model-independent calculation currently exists for nuclei with mass number $A>3$, light nuclei of higher masses are still advantageous in real Compton scattering experiments. Their cross sections are much larger both due to a higher atomic number $Z$ compared to the deuteron, and due to the fact that the meson-exchange currents play a larger role. In particular, $^4$He is a favorable candidate among isoscalar targets because, unlike $^6$Li, its description is well within the reach of modern high-accuracy theoretical approaches. Our data will show that the cross section of Compton scattering from $^4$He is approximately a factor of 6 to 8 larger than that from the deuteron. This substantial enhancement in the cross section enables high statistics measurements. Besides, this confirms the idea that Compton scattering cross sections do not only scale with $Z$ but are also sensitive to the amount of nuclear binding. The Compton scattering cross section, in turn, is related to the number of charged meson-exchange pairs in the target nucleus, see Ref.~\cite{Margaryan18} for details. Additionally, because the first inelastic channel is the $^4$He($\gamma,p$)$^3$H reaction with a $Q$ value of 19.8\,MeV and there are no bound excited states below that energy, elastic Compton scattering from $^4$He can be more easily distinguished from inelastic scattering compared to the deuteron with 2.2\,MeV binding energy. Now that the theory for $^3$He has been explored extensively~\cite{Margaryan18,Shukla:2018rzp,Shukla:2008zc}, one can expect a full theoretical calculation of Compton scattering from $^4$He as a next step. The first precise measurement of the $^4$He Compton scattering cross section was successfully performed at the High Intensity $\gamma$-ray Source (HI$\gamma$S) facility at an incident photon energy of 61\,MeV with high statistical accuracy and well-controlled systematic uncertainties~\cite{Sikora17}. A measurement at a higher photon energy is then motivated in order to obtain greater sensitivity to nucleon polarizabilities and to stimulate the development of EFT calculations of $^4$He Compton scattering to fully interpret the data.

In this paper, we report a new high-precision measurement of the cross section of elastic Compton scattering from $^4$He at a weighted mean incident photon beam energy of 81.3\,MeV. The results are compared to the previous $^4$He Compton scattering data and are discussed in the context of their significance to the extraction of the nucleon polarizabilities.

\section{Experimental Setup}
\label{sec:experiment}
The experiment was performed at HI$\gamma$S at the Triangle Universities Nuclear Laboratory (TUNL)~\cite{Weller09}. The HI$\gamma$S facility utilizes a storage-ring based free-electron laser (FEL) to produce intense, quasi-monoenergetic, and nearly 100\% circularly and linearly polarized $\gamma$-ray beams via Compton backscattering~\cite{Yan:2019bru,Wu:2015hta,Wu:2006zzc}. The $\gamma$-ray beam pulses have a width of about 300\,ps FWHM and are separated by 179\,ns. These features of the HI$\gamma$S photon beam lead to detector energy spectra which are much cleaner and simpler to interpret compared to Compton scattering experiments that use tagged bremsstrahlung beams.

The $\gamma$-ray beam was collimated by a lead collimator with a circular opening of 25.4\,mm diameter located 52.8\,m downstream from the electron-photon collision point. The $\gamma$-ray beam energy was determined from the set-point energy of the storage ring and the measured wavelength of the FEL beam. In this experiment, the calculated energy spectrum of the $\gamma$-ray beam incident on the liquid $^4$He target was peaked at around 85\,MeV with an estimated rms uncertainty of about 1\%~\cite{WYing}. In our experiment an accurate determination of the energy distribution of the $\gamma$-ray beam at the lower energy end was not possible due to the insertion of a set of apertured copper absorbers inside the FEL cavity, which preferably attenuated low energy $\gamma$ rays at larger angles. Instead a study of the beam energy profile was performed to obtain the weighted mean energy of the incident $\gamma$-ray beams. The weighted mean energy was the average energy weighted by the numbers of photons with different energies in the beam. Details of the determination of the weighted mean beam energy are discussed in Sec.~\ref{sec:analysis}. The $\gamma$-ray beam flux was continuously monitored by a system composed of five thin plastic scintillators with an aluminum radiator inserted after the second scintillator~\cite{Pywell09}. This system was located about 70\,cm downstream from the end of the collimator and about 12\,m upstream of the target. The charged particles produced in the radiator were detected to determine the photon beam intensity. The efficiency of the beam flux monitor was measured at low photon fluxes using a large NaI(Tl) detector located downstream of the target and was corrected for multiple hits at high photon rates~\cite{Pywell09}. For the present experiment, the on-target intensity of the circularly polarized photon beams was $\approx10^7\,\gamma/s$.

The beam was scattered from a cryogenic liquid $^4$He target~\cite{Kendellen16}. The near-cylindrical target cell was 20\,cm in length and approximately 4\,cm in diameter. The walls and end windows of the cell were made from 0.125-mm-thick Kapton foil. The cylindrical axis of the target cell was aligned along the beam axis. The target cell was located inside an aluminum vacuum can and two aluminum radiation shields. Two 0.125-mm-thick Kapton windows were installed on the vacuum can, and gaps were cut in the aluminum shields to allow the photon beam to enter and exit the cryogenic target. The liquid $^4$He target was maintained at 3.4\,K with a target thickness of $(4.17\pm0.04) \times 10^{23}$\,nuclei/cm$^2$. During production runs, the cell was periodically emptied for background measurements.

The Compton scattered photons were detected by an array of eight NaI(Tl) detectors placed at scattering angles of $\theta = 55\degree$, 90$\degree$, and 125$\degree$ in the laboratory frame. Each detector consisted of a cylindrical NaI(Tl) core of diameter 25.4\,cm surrounded by eight annular segments of 7.5-cm-thick NaI(Tl) crystals. The lengths of the core NaI crystals ranged from 25.4 to 30.5\,cm. The annular segments were used as an anticoincidence shield to veto the cosmic-ray background. A 15-cm-thick lead collimator was installed at the front face of each detector to define the acceptance cone. The conical aperture of the lead shield was filled with boron-doped paraffin wax to reduce neutron background.

The layout of the detectors and the cryogenic target is shown in Fig.~\ref{fig:Setup}. For this experiment, five detectors (one at $\theta = 55\degree$, two at $\theta = 90\degree$, the other two at $\theta = 125\degree$) were placed on the tables with the axes of the detectors and the target aligned in the same horizontal plane. The other three detectors were placed beneath the beam axis pointing towards the target, located at $\theta = 55\degree$, $90\degree$, and $125\degree$, respectively. The geometry of the experimental apparatus was surveyed to a precision of 0.5\,mm and the results were incorporated into a \textsc{geant4}\xspace~\cite{Agostinelli02} simulation to determine the effective solid angle of each detector. The effective solid angle accounts for the geometric effects due to the extended target and the finite acceptance of the detectors, as well as the attenuation of scattered photons in the target cell and the cryostat. With the front faces of the detector collimator apertures placed about 58\,cm from the target center, the simulated effective solid angles ranged from 63.4 to 66.9\,msr.

\begin{figure}[!htb]
\begin{center}
\includegraphics[width=1\columnwidth]{target_hinda.pdf}
\caption{(Color online) Schematic of the experimental apparatus showing the layout of the cryogenic target and the array of NaI(Tl) detectors. The photon beam is incident from the lower left side of the figure. The target cell is contained inside the aluminum vacuum can. \label{fig:Setup}}
\end{center}
\end{figure}

\section{Data Analysis}
\label{sec:analysis}
Figure~\ref{fig:Daq} illustrates the simplified flow chart of the data acquisition system for this experiment. One copy of the core signal was recorded by the digitizer, while a second copy was used to generate the trigger for the data acquisition system. After passing a hardware threshold of about 10\,MeV, which was set using a constant fraction discriminator (CFD), a logical OR of all core signals was formed to trigger the digitizer. For each event trigger, in addition to recording the waveform of the signal from the core detector that generated the trigger, the waveform of the combined signal from the eight NaI shield segments associated with this core NaI detector and the time of flight (TOF) of the detected $\gamma$-ray events produced by the signal from a time-to-amplitude converter (TAC) were digitized. The TOF was defined as the time difference between an event trigger and the next reference signal of the electron beam pulse from the accelerator every 179\,ns. The energy deposition in the core and shield detectors was extracted from the integral of the pulse shape, while the TOF was obtained from the peak-sensed amplitude of the waveform from the TAC.

\begin{figure}[!htp]
\includegraphics[width=1\columnwidth]{daq.pdf}
\caption{(Color online) Simplified diagram of the data acquisition system. \label{fig:Daq}}
\end{figure}

\begin{figure}[!htp]
\includegraphics[angle=0,width=1\columnwidth]{shield2d.pdf}
\caption{2D spectrum showing energy deposition in the shield detector versus energy deposition in the core detector. An apparent gap around the dashed line is observed between the cosmic-ray events (above the dashed line) and the Compton-scattering events (below the dashed line). The shield energy cut is placed at the dashed line.
\label{fig:shield2d}}
\end{figure}

\begin{figure}[!htp]
\includegraphics[angle=0,width=1\columnwidth]{tof.pdf}
\caption{(Color online) Time-of-flight spectrum showing prompt and random regions with shield-energy cut applied. The bin width is 0.4\,ns.
\label{fig:ToF}}
\end{figure}

Cosmic-ray events were the major source of background in this experiment and could be rejected by employing two methods. First, the energy spectra of the anticoincidence shield detectors were analyzed to suppress such background. Due to the lead collimator in front of each detector, events scattered from the target deposited energy in the shields primarily through electromagnetic shower loss from the core crystal. In contrast, high-energy muons produced by cosmic rays traversed the detector and were minimum ionizing. This significant difference in the shield-energy spectra of the Compton scattered photons and cosmic muons enabled a cut on shield energy (Fig.~\ref{fig:shield2d}) to veto the cosmic-ray background without affecting the Compton scattering events. Secondly, the time structure of the $\gamma$-ray beam produced a clear prompt timing peak (Fig.~\ref{fig:ToF}) for the beam-produced events, allowing for a timing cut to select beam-related scattering events. The shield-energy cut and timing cut together removed over 99\% of the cosmic-ray events within the region of interest (ROI) in the energy spectrum. The background from time-uncorrelated (random) events appeared as a uniform distribution in the TOF spectrum in Fig.~\ref{fig:ToF}. After applying both cuts, the remaining background from residual random events was removed by sampling the energy spectrum from the random region and subtracting it from the energy spectrum in the prompt region after normalizing to the relative widths of the timing windows. The above analysis was performed on both full- and empty-target data. Typical energy spectra from the analysis at the three scattering angles are shown in Fig.~\ref{fig:FullEmpty}. For each detector, the empty-target energy spectrum was subtracted from the full-target energy spectrum after scaling to the number of incident photons to obtain the final energy spectrum.

\begin{figure}[!htp]
\includegraphics[width=1\columnwidth]{full_empty.pdf}
\caption{(Color online) Representative energy spectra for full (closed dot) and empty (open dot) targets at $\theta$ = 55$\degree$, 90$\degree$, and 125$\degree$. The bin width is 0.8\,MeV. The empty-target spectra have been normalized to the number of incident photons for the full-target spectra.
\label{fig:FullEmpty}}
\end{figure}

\begin{figure}[!htp]
\includegraphics[width=1\columnwidth]{lineshape.pdf}
\caption{(Color online) Representative final energy spectra at $\theta$ = 55$\degree$, 90$\degree$, and 125$\degree$ with empty-target events removed. The bin width is 0.8\,MeV. The fit to the data (dashed curve) consists of the electromagnetic background (dot-dashed curve) and the {\textsc{geant4}\xspace} simulated detector response function (solid curve). At $\theta$ = 55$\degree$ and 90$\degree$, the backgrounds are the sum of an exponential low-energy contribution accounting for the atomic scattering and a constant background resulting from the electron-beam-induced bremsstrahlung photons. At $\theta$ = 125$\degree$, the background is free from the exponential low-energy component and therefore includes the constant contribution only. The vertical dashed lines indicate the ROI used to obtain the yield at each angle.
\label{fig:FitSpectra}}
\end{figure}

\begin{figure}[!htb]
\begin{center}
\includegraphics[angle=0,width=1\columnwidth]{beam_profile.pdf}
\caption{The reconstructed effective beam energy profile on the target for a $\gamma$-ray beam produced by an electron beam energy of 975\,MeV in the storage ring and a laser beam of about 192 nm in the FEL optical cavity. This energy distribution is peaked at 85.1\,MeV with a FWHM of 5.5\,MeV (6.5\%). The weighted mean value is 81.3\,MeV.
\label{fig:Beam}}
\end{center}
\end{figure}

\begin{figure}[!htp]
\includegraphics[angle=0,width=1\columnwidth]{cross_section.pdf}
\caption{(Color online) Compton scattering cross sections of $^4$He reported in the current work (circles) compared to the results from Lund (squares, $E_\gamma=$ 87\,MeV)~\cite{Fuhrberg95} and the previous measurement at HI$\gamma$S (triangles, $E_\gamma=$ 61\,MeV)~\cite{Sikora17}. The error bars shown are the statistical and systematic uncertainties added in quadrature.
\label{fig:CrossSections}}
\end{figure}

Typical final energy spectra at the three scattering angles are shown in Fig.~\ref{fig:FitSpectra}. The final energy spectrum of each detector was fitted with the detector response function obtained from the aforementioned {\textsc{geant4}\xspace} simulation of the full experimental apparatus. Photons were generated from the target cell and the energy deposited in the detectors was recorded. The energy of the simulated scattered photon $E_\gamma^{\prime}$ was
\begin{equation}
     E_\gamma^{\prime} = \frac{E_\gamma}{1 + \frac{E_\gamma}{M}(1-\cos\theta)},
\end{equation}
where $E_\gamma$ is the incident photon energy sampled from the effective beam energy profile, $\theta$ is the laboratory scattering angle, and $M$ is the mass of the $^4$He nucleus. The simulated detector response function was fitted to the final energy spectrum with a Gaussian smearing convolution accounting for the intrinsic detector resolution. Due to the lack of a direct method to determine the beam energy profile in the entire distribution range, the effective beam energy profile was determined using the following strategy. A series of samples of beam energy profiles were calculated with a set of beam radii on the target, corresponding to various energy spreads. Each sample was incorporated into the aforementioned simulation to obtain the corresponding detector response function. As the detector response function was fitted to data, the resulting fitting parameter, particularly the width of the Gaussian smearing function representing the intrinsic detector resolution, was evaluated. Only those samples resulting in physically reasonable values of the fitting parameter were taken into account to estimate the weighted mean beam energy, which gave a range of 80.2\,MeV to 84.0\,MeV among the samples. The effective beam energy profile used in the cross-section extraction is shown in Fig.~\ref{fig:Beam} with a FWHM of 5.5\,MeV (6.5\%) and a weighted mean energy value of 81.3\,MeV. A flat background resulting from the scattering of bremsstrahlung photons produced from the 1\,GeV electrons in the storage ring was fitted simultaneously to the final energy spectrum. For the forward-angle detectors, the background from atomic Compton scattering was prominent in the low-energy region and was fitted with an exponential function in addition to the flat background. Typical line-shape fitting for the final energy spectra at forward and backward angles is shown in Fig.~\ref{fig:FitSpectra}. The fitted backgrounds were subtracted from the final energy spectrum, and the number of events was counted within the ROI in the elastic peak as indicated by the vertical dashed lines in Fig.~\ref{fig:FitSpectra}. The yield was extracted by scaling the summed number with an efficiency factor defined as the fraction of the fitted response function within the ROI. The aforementioned shield-energy and timing cuts were chosen such that the maximum yield was obtained while the uncertainty of the yield was minimized. As the lowest-energy breakup reaction $^4$He($\gamma,p$)$^3$H is 19.8\,MeV away from the elastic channel, the inelastic contribution is not expected to contaminate the ROI. The yield was corrected for the absorption of the incident photon beam in the target and normalized to the number of incident photons, target thickness, and effective solid angles to calculate the differential cross sections.

The systematic uncertainties in this experiment were grouped into two categories and are listed in Table~\ref{tab:Systematic}. The first was the point-to-point systematic uncertainty that varied from each datum. This type of uncertainty reflected the effects of placing cuts in the energy and timing spectra of each detector in yield extraction. Values of the point-to-point uncertainties were determined by slightly varying the boundaries of ROI, the windows for the line-shape fitting, the timing cut, and the shield-energy cut. The contribution from the last item was negligible compared to the others. The second type was an overall normalization uncertainty that applied equally to all data. This overall normalization uncertainty included the contributions from the number of incident photons and the target thickness. The uncertainty from using different effective beam profiles from the selected samples was found negligible. Also, the uncertainty from the effective solid angles was evaluated and found negligible due to the high precision in the geometry survey of the experimental apparatus and the small statistical uncertainty of the simulation of the detector system. The contributions of the systematic uncertainties were summed in quadrature to obtain the total systematic uncertainty.

\begin{table}[]
\begin{center}
\caption{Systematic uncertainties.}
\begin{tabular*}{1\columnwidth}{@{\extracolsep{\fill}} l l r }
\hline\hline\noalign{\smallskip}
Type            & Source                        & Value \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Point-to-point  & Timing cut            & 0.3\%--1.7\% \\
                & Line-shape fit window & 0.4\%--1.9\% \\
                & ROI                   & 0.3\%--1.3\% \\
\noalign{\smallskip}\noalign{\smallskip}
Normalization   & Number of incident photons    & 2.0\% \\
                & Target thickness              & 1.0\% \\
\noalign{\smallskip}\noalign{\smallskip}
Total           &                               & 2.5\%--3.5\%  \\
\noalign{\smallskip}\hline\hline
\end{tabular*}
\label{tab:Systematic}
\end{center}
\end{table}

\begin{table}[!htp]
\begin{center}
\caption{The Compton scattering cross section of $^4$He at a weighted mean beam energy of 81.3\,MeV measured by the eight NaI(Tl) detectors used in our experiment. The last three columns list the statistical, point-to-point systematic, and total systematic uncertainties.}
\begin{tabular*}{1\columnwidth}{@{\extracolsep{\fill}} l c c c c c}
\hline\hline\noalign{\smallskip}
$\theta_{\text{Lab}}$  &$\phi$ & $d\sigma/d\Omega$ & Stat & Point-to-point &Total Syst \\
\noalign{\smallskip}
 &  & (nb/sr) & (nb/sr) &Syst (nb/sr) &(nb/sr)\\
\noalign{\smallskip}\hline\noalign{\smallskip}
 $55\degree$    & $0\degree$    & 75.1  &$\pm2.6$   &$\pm1.1$  &$\pm2.0$ \\
 $55\degree$    & $270\degree$  & 81.6  &$\pm2.8$   &$\pm2.2$  &$\pm2.9$ \\
\noalign{\smallskip}\noalign{\smallskip}
 $90\degree$    & $0\degree$    & 58.8  &$\pm2.2$   &$\pm1.0$  &$\pm1.6$ \\
 $90\degree$    & $180\degree$  & 66.7  &$\pm2.7$   &$\pm1.0$  &$\pm1.8$ \\
 $90\degree$    & $270\degree$  & 61.8  &$\pm2.2$   &$\pm0.7$  &$\pm1.6$ \\
\noalign{\smallskip}\noalign{\smallskip}
 $125\degree$   & $0\degree$    & 90.6  &$\pm2.6$   &$\pm1.2$  &$\pm2.4$ \\
 $125\degree$   & $180\degree$  & 97.2  &$\pm2.6$   &$\pm1.2$  &$\pm2.5$ \\
 $125\degree$   & $270\degree$  & 102.1  &$\pm2.4$   &$\pm1.1$  &$\pm2.5$ \\
\noalign{\smallskip}\hline\hline
\end{tabular*}
\label{tab:CrossSections}
\end{center}
\end{table}

\begin{table}[!htp]
\begin{center}
\caption{The averaged Compton scattering cross section of $^4$He measured at weighted mean beam energy 81.3\,MeV at the three angles of our experimental setup. These data are plotted in Fig.~\ref{fig:CrossSections}. The third column lists the statistical uncertainties. The last two columns list the point-to-point and total systematic uncertainties.}
\begin{tabular*}{1\columnwidth}{@{\extracolsep{\fill}} l c c c c}
\hline\hline\noalign{\smallskip}
$\theta_{\text{Lab}}$  & $d\sigma/d\Omega$ & Stat & Point-to-point & Total Syst \\
\noalign{\smallskip}
 & (nb/sr) & (nb/sr) &Syst (nb/sr) &(nb/sr)\\
\noalign{\smallskip}\hline\noalign{\smallskip}
 $55\degree$ &   78.0 &$\pm1.9$ &$\pm1.7$ &$\pm2.4$\\
 $90\degree$ &   61.9 &$\pm1.3$ &$\pm0.9$ &$\pm1.6$\\
 $125\degree$ &  97.0 &$\pm1.5$ &$\pm1.2$ &$\pm2.5$\\
\noalign{\smallskip}\hline\hline
\end{tabular*}
\label{tab:Results}
\end{center}
\end{table}

\section{Results and Discussion}
\label{sec:results}
The differential cross section extracted for each detector is listed in Table~\ref{tab:CrossSections}. For different detectors at the same angle $\theta$, the cross sections are overall in good agreement with each other, although spreads among different azimuthal angles are more pronounced at $125\degree$. Part of the spreads can be accounted for once systematic uncertainties are taken into account. However, systematic aspects due to detection variations among detectors are difficult to take into account in our analysis but can likely contribute to the observed spreads. The cross section value at each scattering angle in the end is assigned as the weighted average by the statistical uncertainties. The reduced $\chi^{2}$ is calculated at each scattering angle with respect to the weighted average value. The calculated reduced $\chi^{2}$ values are $2.07$, $2.17$, and $4.36$ at $\theta = 55\degree$, $90\degree$, and $125\degree$, respectively. The final results are plotted in Fig.~\ref{fig:CrossSections} and listed in Table~\ref{tab:Results}. The elastic Compton scattering data from Lund~\cite{Fuhrberg95} at an incident photon energy of 87\,MeV and the previous HI$\gamma$S measurement~\cite{Sikora17} at 61\,MeV are also shown in Fig.~\ref{fig:CrossSections}. The results of this work are in good agreement with the Lund results. Particularly, the present data follow the same fore-aft asymmetry in the angular distribution as the Lund data, with a strong backward peaking that is evident in both data sets. This asymmetry is distinct from the 61-MeV HI$\gamma$S data and indicates a strong sensitivity to sub-nuclear effects, including the nucleon polarizabilities. An accurate theoretical calculation of the reaction, currently lacking, is needed to explain this behavior as well as to extract the values for the isoscalar polarizabilities from these data. Such a calculation has already been done for Compton scattering from $^3$He using an EFT framework~\cite{Margaryan18,Shukla:2018rzp,Shukla:2008zc}, therefore the prospects for such a treatment for $^4$He are very promising. The high statistical accuracy of the present work and the previous HI$\gamma$S measurement at 61\,MeV provide a strong motivation for further theoretical work on $^4$He in order to extract the neutron polarizabilities with a precision that is difficult to achieve from deuterium experiments.

\section{Summary}
\label{sec:summary}
Elastic Compton scattering from $^4$He provides a complementary approach to deuteron experiments that allows for the extraction of the nucleon polarizabilities. To this end, a new high-precision measurement of the cross section of Compton scattering from $^4$He at 81.3\,MeV has been performed at HI$\gamma$S. While the results exhibit a behavior similar to that seen in previously reported data, this experimental work has achieved an unprecedented level of accuracy. This work, together with the HI$\gamma$S measurement at 61\,MeV, is expected to strongly spur the development of a rigorous theoretical treatment to interpret the $^4$He Compton scattering data in order to extract the polarizabilities of the proton and neutron with high accuracy.

\acknowledgments{
This work is funded in part by the US Department of Energy under Contracts No.~DE-FG02-03ER41231, DE-FG02-97ER41033, DE-FG02-97ER41041, DE-FG02-97ER41046, DE-FG02-97ER41042, DE-SC0005367, DE-SC0015393, DE-SC0016581, and DE-SC0016656, National Science Foundation Grants No.~NSF-PHY-0619183, NSF-PHY-1309130, and NSF-PHY-1714833, and funds from the Dean of the Columbian College of Arts and Sciences at The George Washington University, and its Vice-President for Research. We acknowledge the financial support of the Natural Sciences and Engineering Research Council of Canada and the support of Eugen-Merzbacher Fellowship. We acknowledge the support of the HI$\gamma$S accelerator scientists, engineers, and operators for assisting with the experimental setup, tuning up the accelerator system for high-energy operation, and for the high-quality production of the $\gamma$-ray beam.
}