Proof of A Log-Sobolev Inequality for the Galerkin Wick-Ordered $Phi^4$ Gibbs Measure on the Torus at Temperatures above the Semiconvexity Constant

Each result cited is universally quantified over the data in its own statement. Elementary real arithmetic and order (The Real Numbers: Standing Notation and Background §background) are used without citation. In this proof the letter $\kappa$ keeps its meaning in The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §data (the enumeration of $\mathbb{Z}^{n}$), and $n$ is the dimension of the torus; the positive constant written $\kappa$ in Log-Sobolev and Talagrand Inequalities for the Gibbs Measure of a Semiconvex Cylindrical Potential below the Critical Curvature (Bakry-Emery) is called the constant of that item, and the potential and semiconvexity constant written $V$ and $K$ there are called its potential and its constant $K$.

Step 1 (The setting). By The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §gaussian, the notation of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation is in force with the real Hilbert space $X=H^{-m}(\mathbb{T}^{n})$, its orthonormal basis $(e_{j})_{j\in\mathbb{N}}$, the weight sequence $a$ with bound $\bar{a}=1$ and the variance sequence $c$ of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §data, the reference measure being $\rho=\gamma_{c}$. The setting Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation fixes notation only. Hence the background clause Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §background of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, which is the setting of Log-Sobolev and Talagrand Inequalities for the Gibbs Measure of a Semiconvex Cylindrical Potential below the Critical Curvature (Bakry-Emery), is instantiated by these data, and the remaining clauses of that setting only name objects built from them. Under this instantiation the objects of the present statement are those of Log-Sobolev and Talagrand Inequalities for the Gibbs Measure of a Semiconvex Cylindrical Potential below the Critical Curvature (Bakry-Emery): relative entropy and $H(\cdot\,|\,\cdot)$ are those of Relative Entropy of Probability Measures §relative-entropy in both (A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian); the Gibbs measure, the relative score with respect to it and the Fisher information relative to it with weights $a$ are those of The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure §gibbs, The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential §score and The Weighted Fisher Information Relative to the Gibbs Measure of an Admissible Cylindrical Potential §information; $\mathcal{F}C^{1}_{b}(X)$ and $\nabla_{a}F$ are those of Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical and The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient (Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §cylindrical); and the entropy is that of The Entropy of a Nonnegative Function with Respect to a Probability Measure §entropy. The indices written $n$ in the coordinate maps $p_{n}$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, in the representations $(n,\psi)$ of Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §representation and in Log-Sobolev and Talagrand Inequalities for the Gibbs Measure of a Semiconvex Cylindrical Potential below the Critical Curvature (Bakry-Emery) are bound variables of those items, ranging over $\mathbb{N}$, and are unaffected by the torus dimension $n$ fixed by the present setting; the letter $N$ denotes the cutoff throughout. Thus Log-Sobolev and Talagrand Inequalities for the Gibbs Measure of a Semiconvex Cylindrical Potential below the Critical Curvature (Bakry-Emery) is applied to the torus data above, as an instance of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation.

Step 2 (The hypotheses of the corollary). Take the constant of Log-Sobolev and Talagrand Inequalities for the Gibbs Measure of a Semiconvex Cylindrical Potential below the Critical Curvature (Bakry-Emery) equal to $1$, which is positive, and satisfies $c_{j}\le1\cdot a_{j}$ for every $j\in\mathbb{N}$ by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §ratio. Take its constant $K$ equal to $K_{N}$ and its potential equal to $V_{N}$: by The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass §admissible, $0\le K_{N}$ and $V_{N}$ is an admissible cylindrical potential with head dimension $d_{N}$, profile $v_{N}$ and semiconvexity constant $K_{N}$, as Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry §admissible requires. The temperature $\beta$ is positive, since $0\le K_{N}<\beta$, and $1\cdot K_{N}=K_{N}<\beta$. So all hypotheses of Log-Sobolev and Talagrand Inequalities for the Gibbs Measure of a Semiconvex Cylindrical Potential below the Critical Curvature (Bakry-Emery) hold, with $\gamma^{V_{N}}_{\beta}$ in place of the Gibbs measure written there, and the constants there become

$$\frac{1\cdot\beta}{2(\beta-1\cdot K_{N})}=\frac{\beta}{2(\beta-K_{N})},\qquad\frac{2\cdot1\cdot\beta}{\beta-1\cdot K_{N}}=\frac{2\beta}{\beta-K_{N}} .$$

Step 3 (Claim 1). Let $\mu\in\mathcal{P}(X)$ have finite relative entropy with respect to $\gamma^{V_{N}}_{\beta}$, a relative score with respect to $\gamma^{V_{N}}_{\beta}$ and finite Fisher information relative to $\gamma^{V_{N}}_{\beta}$ with weights $a$. By Log-Sobolev and Talagrand Inequalities for the Gibbs Measure of a Semiconvex Cylindrical Potential below the Critical Curvature (Bakry-Emery) §entropy and Step 2,

$$H(\mu\,|\,\gamma^{V_{N}}_{\beta})\le\frac{\beta}{2(\beta-K_{N})}\,\mathcal{I}_{a}(\mu\,|\,\gamma^{V_{N}}_{\beta}).$$

Step 4 (Claim 2). Let $F\in\mathcal{F}C^{1}_{b}(X)$. By Log-Sobolev and Talagrand Inequalities for the Gibbs Measure of a Semiconvex Cylindrical Potential below the Critical Curvature (Bakry-Emery) §gross and Step 2, $F^{2}$ and $\phi\circ F^{2}$, with $\phi$ the function of The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm, are bounded and Borel, hence integrable with respect to $\gamma^{V_{N}}_{\beta}$, which is the first assertion of claim 2, so that $\operatorname{Ent}_{\gamma^{V_{N}}_{\beta}}(F^{2})$ is defined, and

$$\operatorname{Ent}_{\gamma^{V_{N}}_{\beta}}(F^{2})\le\frac{2\beta}{\beta-K_{N}}\int_{X}|\nabla_{a}F|_{a}^{2}\,d\gamma^{V_{N}}_{\beta}.$$