Proof of 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

Each result cited is universally quantified over the data in its own statement. Real order and arithmetic are used through the results placed in force by The Real Numbers: Standing Notation and Background §background.

Step 1 (Claim 1). Let $\varsigma$ be an admissible split. We apply The Gibbs Entropy Pair of a K-Semiconvex Potential is Lambda-Displacement Convex with Lambda the Temperature over the Variance-to-Noise Bound Minus K §convex in the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation read as in 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 the space $X$, its basis $(e_{j})$, the noise weights $a$ with bound $1$ and the variance sequence $c^{(\varsigma)}$ in place of $c$ (a variance sequence by 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), so that $\rho=\gamma_{c^{(\varsigma)}}$. Its data are instantiated as follows. $K:=K_{\varsigma}$, the greater of $0$ and $1-\varsigma^{-1}$, hence nonnegative, and $V:=V_{\varsigma}$, which is an admissible cylindrical potential with semiconvexity constant $K_{\varsigma}$ by 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. $\beta:=1$, which is positive, and the positive constant of that lemma (written $\kappa$ there, a letter used here for the enumeration) $:=\bar{r}_{\varsigma}$, which is at least $1$, being the greatest of $1$ and the numbers $r^{(\varsigma)}_{k}$ ($k\in\Gamma_{M}$), hence positive; the required bound $c^{(\varsigma)}_{j}\le\bar{r}_{\varsigma}\,a_{j}$ for every $j\in\mathbb{N}$ is 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. With these data the Gibbs entropy pair of that lemma is the Gibbs entropy pair with potential $V_{\varsigma}$ and temperature $1$ relative to $\gamma_{c^{(\varsigma)}}$, which by 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 is the quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ of the statement. So The Gibbs Entropy Pair of a K-Semiconvex Potential is Lambda-Displacement Convex with Lambda the Temperature over the Variance-to-Noise Bound Minus K §convex shows that this quadruple is $\bigl(\frac{1}{\bar{r}_{\varsigma}}-K_{\varsigma}\bigr)$-displacement convex, in the sense of Lambda-Displacement Convexity of a Noise Penalty Pair §convex read with $c^{(\varsigma)}$; as $\frac{1}{\bar{r}_{\varsigma}}=\bar{r}_{\varsigma}^{-1}$, this is claim 1.

Step 2 (Monotonicity in $\lambda$). Let $\varsigma$ be an admissible split, let $\lambda',\lambda''\in\mathbb{R}$ with $\lambda'\le\lambda''$, and suppose the pair is $\lambda''$-displacement convex in the sense of Lambda-Displacement Convexity of a Noise Penalty Pair §convex read with $c^{(\varsigma)}$. Let $\mu\in\mathcal{D}_{\Sigma}$, $\nu\in\mathcal{D}$ and let $\pi\in\Pi^{a}(\mu,\nu)$ be a noise-optimal coupling. Its noise cost $I^{a}(\pi)$ is that of Couplings of Finite Noise Cost and Their Noise Cost §cost, by First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation §couplings, and is a nonnegative real number by Couplings of Finite Noise Cost and Their Noise Cost §cost; hence $\frac{\lambda'}{2}I^{a}(\pi)\le\frac{\lambda''}{2}I^{a}(\pi)$ and

$$\mathcal{E}(\mu)+\mathcal{J}^{a}(\Sigma(\mu),\pi)+\tfrac{\lambda'}{2}I^{a}(\pi)\le\mathcal{E}(\mu)+\mathcal{J}^{a}(\Sigma(\mu),\pi)+\tfrac{\lambda''}{2}I^{a}(\pi)\le\mathcal{E}(\nu).$$

As $\mu$, $\nu$ and $\pi$ were arbitrary, the pair is $\lambda'$-displacement convex.

Step 3 (The split $1+\bar{\mathfrak{q}}$). Let $\bar{\mathfrak{q}}$ be as in claim 2. $\Gamma_{M}$ is nonempty by The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite; for any $k\in\Gamma_{M}$ we have $0<\mathfrak{q}_{k}\le\bar{\mathfrak{q}}$, by The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §shift and the hypothesis, so $0<\bar{\mathfrak{q}}<1$. Put $\varsigma=1+\bar{\mathfrak{q}}$. For every $k\in\Gamma_{M}$, $\mathfrak{q}_{k}\le\bar{\mathfrak{q}}<1+\bar{\mathfrak{q}}=\varsigma$, so $\varsigma$ is an admissible split, as defined in 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. Since $\varsigma>1$, $1-\varsigma^{-1}=1-\frac{1}{1+\bar{\mathfrak{q}}}=\frac{\bar{\mathfrak{q}}}{1+\bar{\mathfrak{q}}}>0$, so $K_{\varsigma}=\frac{\bar{\mathfrak{q}}}{1+\bar{\mathfrak{q}}}$ by its definition in 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.

Step 4 (A bound for $\bar{r}_{\varsigma}$). Let $k\in\Gamma_{M}$. By 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, $r^{(\varsigma)}_{k}$ is positive and $1/r^{(\varsigma)}_{k}=1/\mathfrak{q}_{k}-1/\varsigma$. Since $0<\mathfrak{q}_{k}\le\bar{\mathfrak{q}}$, $1/\mathfrak{q}_{k}\ge1/\bar{\mathfrak{q}}$, so

$$\frac{1}{r^{(\varsigma)}_{k}}\ge\frac{1}{\bar{\mathfrak{q}}}-\frac{1}{1+\bar{\mathfrak{q}}}=\frac{1}{\bar{\mathfrak{q}}(1+\bar{\mathfrak{q}})}>0,$$

and, both sides being positive, $r^{(\varsigma)}_{k}\le\bar{\mathfrak{q}}(1+\bar{\mathfrak{q}})$. Let $\bar{s}$ be the larger of $1$ and $\bar{\mathfrak{q}}(1+\bar{\mathfrak{q}})$. By 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, $\bar{r}_{\varsigma}$ is the greatest entry of the tuple consisting of $1$ followed by the numbers $r^{(\varsigma)}_{k}$ ($k\in\Gamma_{M}$), given by Greatest Element of a Finite Family in a Totally Ordered Set as one of these entries; each entry is at most $\bar{s}$, so $0<\bar{r}_{\varsigma}\le\bar{s}$ and hence $\frac{1}{\bar{r}_{\varsigma}}\ge\frac{1}{\bar{s}}$.

Step 5 (The constant). We show $\frac{1}{\bar{s}}-K_{\varsigma}\ge\lambda(\bar{\mathfrak{q}})$. If $\bar{\mathfrak{q}}(1+\bar{\mathfrak{q}})\le1$, then $\bar{s}=1$ and

$$\frac{1}{\bar{s}}-K_{\varsigma}=1-\frac{\bar{\mathfrak{q}}}{1+\bar{\mathfrak{q}}}=\frac{1}{1+\bar{\mathfrak{q}}}\ge\lambda(\bar{\mathfrak{q}}).$$

If $\bar{\mathfrak{q}}(1+\bar{\mathfrak{q}})>1$, then $\bar{s}=\bar{\mathfrak{q}}(1+\bar{\mathfrak{q}})$ and

$$\frac{1}{\bar{s}}-K_{\varsigma}=\frac{1-\bar{\mathfrak{q}}^{2}}{\bar{\mathfrak{q}}(1+\bar{\mathfrak{q}})}=\frac{(1-\bar{\mathfrak{q}})(1+\bar{\mathfrak{q}})}{\bar{\mathfrak{q}}(1+\bar{\mathfrak{q}})}=\frac{1-\bar{\mathfrak{q}}}{\bar{\mathfrak{q}}}\ge\lambda(\bar{\mathfrak{q}}).$$

In both cases the last inequality holds because $\lambda(\bar{\mathfrak{q}})$ is the smaller of the two numbers in its definition. With Step 4, $\frac{1}{\bar{r}_{\varsigma}}-K_{\varsigma}\ge\frac{1}{\bar{s}}-K_{\varsigma}\ge\lambda(\bar{\mathfrak{q}})$.

Step 6 (Claim 2). By Step 1 for the admissible split $\varsigma=1+\bar{\mathfrak{q}}$ of Step 3, the pair is $\bigl(\frac{1}{\bar{r}_{\varsigma}}-K_{\varsigma}\bigr)$-displacement convex, the notion being read with $c^{(\varsigma)}$; by Step 5 and Step 2 (with $\lambda'=\lambda(\bar{\mathfrak{q}})$ and $\lambda''=\frac{1}{\bar{r}_{\varsigma}}-K_{\varsigma}$) it is $\lambda(\bar{\mathfrak{q}})$-displacement convex. Since $\bar{\mathfrak{q}}<1$ and $0<\bar{\mathfrak{q}}$, both $\frac{1}{1+\bar{\mathfrak{q}}}$ and $\frac{1-\bar{\mathfrak{q}}}{\bar{\mathfrak{q}}}$ are positive, so $0<\lambda(\bar{\mathfrak{q}})$; Step 2 with $\lambda'=0$ and $\lambda''=\lambda(\bar{\mathfrak{q}})$ shows the pair is $0$-displacement convex, that is, displacement convex. Finally, $\lambda(\bar{\mathfrak{q}})$ is given by its defining formula in terms of $\bar{\mathfrak{q}}$ alone, in which $M$, $\mathfrak{j}$ and $\eta$ do not occur; so it depends on $\bar{\mathfrak{q}}$ only, which is the last sentence of claim 2.

Step 7 (The reading does not depend on the split). The readings of Lambda-Displacement Convexity of a Noise Penalty Pair with different admissible splits $\varsigma$ and $\varsigma'$ agree for this pair, so that claims 1 and 2 hold in every such reading. Indeed, the quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ and the set $\mathcal{P}^{a}_{\gamma_{c^{(\varsigma)}}}=\mathcal{P}^{a}_{\gamma_{c^{(\varsigma')}}}$ are the same for both, by 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, and the quadruple is a noise penalty pair on that set by the statement. The remaining ingredients of the defining inequality of Lambda-Displacement Convexity of a Noise Penalty Pair §convex are formed in the notation of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation from the noise weights $a$, which are the same in both readings, and the measures involved, without the reference measure: the sets $\Pi^{a}(\mu,\nu)$ and the noise cost $I^{a}$ of Couplings of Finite Noise Cost and Their Noise Cost §couplings and Couplings of Finite Noise Cost and Their Noise Cost §cost; noise-connectedness of $(\mu,\nu)$ in the sense of Couplings of Finite Noise Cost and Their Noise Cost §connected, which is nonemptiness of $\Pi^{a}(\mu,\nu)$ and does not involve the reference measure (it is obtained via The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected from $\mu,\nu\in\mathcal{P}^{a}_{\gamma_{c^{(\varsigma)}}}$, a set that is the same in both readings, as just noted); the noise tangent space $T^{a}_{\mu}$ of The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §tangent, the closure in $L^{2}(\mu;X^{a})$ of the classes of noise gradients of bounded $C^{1}$ cylindrical functions, formed from $a$ and $\mu$ alone; the distance $W_{a}$ of The Noise Wasserstein Distance §distance and hence the noise-optimal couplings of Noise-Optimal Couplings §optimal; and the space $L^{2}(\mu;X^{a})$ and the noise displacement pairing $\mathcal{J}^{a}(\Sigma(\mu),\pi)$ of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §pairing.