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

In the setting of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation, so that $X=H^{-m}(\mathbb{T}^{n})$ with orthonormal basis $(e_{j})_{j\in\mathbb{N}}$, noise weights $a_{j}$, variances $c_{j}$ and reference measure $\gamma_{c}$, let $x_{j}$ be the $j$-th coordinate of $x\in X$ and $p_{d}:X\to\mathbb{R}^{d}$ ($d\in\mathbb{N}$) the coordinate map, as in A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §background; $X^{a}$ and $|\cdot|_{a}$ are the noise space and noise norm, described 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 §noise-space. Let $N\in\mathbb{N}$, and let $\varkappa,\mathfrak{m}\in\mathbb{R}$ satisfy $0<\varkappa$. The notation of The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus is used for this cutoff, coupling and mass: the maps $\psi_{k}$ and numbers $\mu_{k}^{-1}$ ($k\in\mathbb{Z}^{n}$), the measure $\lambda_{Q}$ on the cell $Q$, the numbers $a_{j}^{-1/2}$ and the value $x(k)$ of the coefficient family $x\in X$ at $k$, as fixed in its preamble; the cube $\Gamma_{N}$; the Galerkin head dimension $d_{N}$, with the indicator $\chi_{N}$, the set $J_{N}$ of active indices (those $j\in[d_{N}]$ with $\kappa(j)\in\Gamma_{N}$) and the convention $\sum_{j\in J_{N}}t_{j}=\sum_{j=1}^{d_{N}}\chi_{N}(j)\,t_{j}$ of that clause; the Galerkin fields $\Phi_{N}[x]$ ($x\in X$) and $\varphi_{N}[u]$ ($u\in\mathbb{R}^{d_{N}}$); the Wick constant $C_{N}$; the Wick polynomial $f_{N}$; and the Galerkin Wick-ordered $\phi^{4}$ potential $V_{N}:X\to\mathbb{R}$ with its profile $v_{N}:\mathbb{R}^{d_{N}}\to\mathbb{R}$. For $N'\in\mathbb{N}$, $C_{N'}$ is the Wick constant at cutoff $N'$. Sums over the finite set $\Gamma_{N}$ are sums over a finite index set. Functions of class $C^{2}$ on $\mathbb{R}^{d_{N}}$ are those of C^k Maps on a Euclidean Open Set, with the partial derivatives $\partial_{j}v$ and $\partial_{j}\partial_{k}v$; $\max\{s,s'\}$ is the maximum of $s,s'\in\mathbb{R}$. Put

$$K_{N}=\max\{3\varkappa C_{N}-\mathfrak{m},0\}.$$

1. (The Wick constant)¶ $0<C_{N}$, and

$$C_{N}=\sum_{k\in\Gamma_{N}}\mu_{k}^{-1}=\sum_{j\in J_{N}}\frac{c_{j}}{a_{j}} .$$

For every $z\in\mathbb{R}^{n}$,

$$\sum_{j\in J_{N}}\mu_{\kappa(j)}^{-1}\,\psi_{\kappa(j)}(z)^{2}=C_{N};$$

moreover the functions $x\mapsto\Phi_{N}[x](z)$ and $x\mapsto\Phi_{N}[x](z)^{2}$ on $X$ are integrable with respect to $\gamma_{c}$, with

$$\int_{X}\Phi_{N}[x](z)\,\gamma_{c}(dx)=0,\qquad\int_{X}\Phi_{N}[x](z)^{2}\,\gamma_{c}(dx)=C_{N}.$$

2. (Coordinates and the noise norm)¶ For every $x\in X$ and $j\in[d_{N}]$, $x(\kappa(j))=a_{j}^{-1/2}x_{j}$; hence $\Phi_{N}[x]=\varphi_{N}[p_{d_{N}}(x)]$ for every $x\in X$, and $V_{N}=v_{N}\circ p_{d_{N}}$. The profile depends only on the active coordinates: if $u,u'\in\mathbb{R}^{d_{N}}$ satisfy $u_{j}=u'_{j}$ for every $j\in J_{N}$, then $\varphi_{N}[u]=\varphi_{N}[u']$ and $v_{N}(u)=v_{N}(u')$. For every $h\in\mathbb{R}^{d_{N}}$ the class $U=[\,\varphi_{N}[h]|_{Q}\,]\in L^{2}(\mathbb{T}^{n})$ satisfies $\hat{U}=\sum_{j\in J_{N}}h_{j}e_{j}$, and

$$\int_{Q}\varphi_{N}[h](z)^{2}\,\lambda_{Q}(dz)=\lVert U\rVert_{L^{2}}^{2}=\Bigl|\sum_{j\in J_{N}}h_{j}e_{j}\Bigr|_{a}^{2}=\sum_{j\in J_{N}}\frac{h_{j}^{2}}{a_{j}}\ \le\ \sum_{k=1}^{d_{N}}\frac{h_{k}^{2}}{a_{k}} .$$

3. (The Hessian)¶ $v_{N}$ is of class $C^{2}$ on $\mathbb{R}^{d_{N}}$. For every $j\in[d_{N}]$ with $\kappa(j)\notin\Gamma_{N}$ and every $k\in[d_{N}]$, the partial derivatives $\partial_{j}v_{N}$, $\partial_{j}\partial_{k}v_{N}$ and $\partial_{k}\partial_{j}v_{N}$ vanish identically on $\mathbb{R}^{d_{N}}$. For all $u,h\in\mathbb{R}^{d_{N}}$,

$$\sum_{j=1}^{d_{N}}\sum_{k=1}^{d_{N}}\partial_{j}\partial_{k}v_{N}(u)\,h_{j}h_{k}=\int_{Q}\bigl(3\varkappa\,\varphi_{N}[u](z)^{2}+\mathfrak{m}-3\varkappa C_{N}\bigr)\,\varphi_{N}[h](z)^{2}\,\lambda_{Q}(dz)\ \ge\ -K_{N}\sum_{j\in J_{N}}\frac{h_{j}^{2}}{a_{j}}\ \ge\ -K_{N}\sum_{k=1}^{d_{N}}\frac{h_{k}^{2}}{a_{k}} .$$

4. (Admissibility)¶ $0\le K_{N}$, and $V_{N}$ is an admissible cylindrical potential with head dimension $d_{N}$, profile $v_{N}$ and semiconvexity constant $K_{N}$ in the noise norm.

5. (The noise gradient)¶ For every $x\in X$, the noise gradient of $V_{N}$ is

$$\nabla_{a}V_{N}(x)=\sum_{j\in J_{N}}\Bigl(\int_{Q}\Bigl(\varkappa\bigl(\Phi_{N}[x](z)^{3}-3C_{N}\Phi_{N}[x](z)\bigr)+\mathfrak{m}\,\Phi_{N}[x](z)\Bigr)\psi_{\kappa(j)}(z)\,\lambda_{Q}(dz)\Bigr)\hat{E}_{\kappa(j)} .$$

6. (The Wick constant diverges)¶ Suppose $2\le n$. Then for every $M\in\mathbb{R}$ there is $N_{0}\in\mathbb{N}$ with $M<C_{N'}$ for every $N'\in\mathbb{N}$ with $N_{0}\le N'$.