The Gibbs Entropy Pair of the Hubbard-Stratonovich Ising Field is Lambda-Displacement Convex, with Lambda Uniform in the Cutoff when the Shifted Symbol Stays below One

In the situation of The Hubbard-Stratonovich Field Law is the Gibbs Measure of an Admissible Cylindrical Potential for Every Admissible Split of Its Gaussian, and the Gibbs Entropy Pair Does Not Depend on the Split, with $M\in\mathbb{N}$, the reflection-symmetric periodic pair interaction $\mathfrak{j}$ of side $L=2M+1$ with symbol $\hat{J}$ and the shift $\eta\in\mathbb{R}$ for $\mathfrak{j}$ as there; $\mathfrak{q}_{k}=\hat{J}(k)+\eta$ is positive for every $k$ in the cube $\Gamma_{M}$, because $\eta$ is a shift for $\mathfrak{j}$, and no further condition is placed on $M$, $\mathfrak{j}$ or $\eta$. Let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gibbs entropy pair of The Hubbard-Stratonovich Field Law is the Gibbs Measure of an Admissible Cylindrical Potential for Every Admissible Split of Its Gaussian, and the Gibbs Entropy Pair Does Not Depend on the Split §independence, which is the same for every admissible split; for every admissible split $\varsigma$ it is a noise penalty pair on $\mathcal{P}^{a}_{\gamma_{c^{(\varsigma)}}}$ by The Gibbs Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair. For an admissible split $\varsigma$, $\bar{r}_{\varsigma}$ is the constant of The Hubbard-Stratonovich Field Law is the Gibbs Measure of an Admissible Cylindrical Potential for Every Admissible Split of Its Gaussian, and the Gibbs Entropy Pair Does Not Depend on the Split §split and $K_{\varsigma}$ the semiconvexity constant of The Hubbard-Stratonovich Field Law is the Gibbs Measure of an Admissible Cylindrical Potential for Every Admissible Split of Its Gaussian, and the Gibbs Entropy Pair Does Not Depend on the Split §potential. $\lambda$-displacement convexity and displacement convexity are those of Lambda-Displacement Convexity of a Noise Penalty Pair §convex and Lambda-Displacement Convexity of a Noise Penalty Pair §displacement-convex, read in the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation with $c^{(\varsigma)}$ in place of $c$.

1. ($\lambda$-displacement convexity)¶ For every admissible split $\varsigma$, the pair is $\bigl(\bar{r}_{\varsigma}^{-1}-K_{\varsigma}\bigr)$-displacement convex.

2. (High temperature)¶ Let $\bar{\mathfrak{q}}\in\mathbb{R}$ satisfy $\mathfrak{q}_{k}\le\bar{\mathfrak{q}}<1$ for every $k\in\Gamma_{M}$, and put

$$\lambda(\bar{\mathfrak{q}})=\min\Bigl\{\frac{1}{1+\bar{\mathfrak{q}}},\ \frac{1-\bar{\mathfrak{q}}}{\bar{\mathfrak{q}}}\Bigr\},$$

which is meaningful and positive, since $0<\mathfrak{q}_{k}\le\bar{\mathfrak{q}}<1$ for the elements $k$ of the nonempty set $\Gamma_{M}$. Then $1+\bar{\mathfrak{q}}$ is an admissible split, and the pair is $\lambda(\bar{\mathfrak{q}})$-displacement convex, the notion being read with $c^{(1+\bar{\mathfrak{q}})}$ in place of $c$; in particular it is displacement convex, read in the same way. ¶The number $\lambda(\bar{\mathfrak{q}})$ depends on $\bar{\mathfrak{q}}$ only, and not on $M$, $\mathfrak{j}$ or $\eta$.