The Wick Square of the Free Field on the Torus as Diagonal Couplings: Summability in Dimension at Most Three, the Riccati Condition, and the Divergent Wick Constant

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 $n\le m+1$, the weights are $a_{j}=(\rho^{m}_{\kappa(j)})^{2}$ and the variances $c_{j}=(\rho^{m+1}_{\kappa(j)})^{2}$, where $\rho_{k}^{2}=\mu_{k}^{-1}$; let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian entropy pair with temperature $1$, whose hypothesis holds with the constant $1$ 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. For $x\in X$, $x_{j}$ is its $j$-th coordinate as in A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §background, and $\mu_{k}^{m}$ is the natural power; $\mu_{k}^{-1}$ is the multiplicative inverse of $\mu_{k}$, which is positive since $1\le\mu_{k}$, and $\mu_{k}^{-2}$ that of $\mu_{k}^{2}$. Let $w_{0}\in\mathbb{R}$ and let $w=(w_{j})_{j\in\mathbb{N}}$ with $w_{j}=w_{0}\,\mu_{\kappa(j)}^{m}$. Convergent series are those of that definition, and $\min\{s,s'\}$ is the minimum of $s,s'\in\mathbb{R}$.

1. (The couplings are the Wick square)¶ For every $x\in X$ and $j\in\mathbb{N}$, $w_{j}\,x_{j}^{2}=w_{0}\,x(\kappa(j))^{2}$, where $x(k)$ is the value at $k\in\mathbb{Z}^{n}$ of the coefficient family $x$; moreover $w_{j}c_{j}=w_{0}\,\mu_{\kappa(j)}^{-1}$ and $|w_{j}|\,c_{j}\le|w_{0}|$.

2. (Summability)¶ Suppose $n\le3$. Then the series $\sum_{j=1}^{\infty}|w_{j}|\,c_{j}^{2}/a_{j}=\sum_{j=1}^{\infty}|w_{0}|\,\mu_{\kappa(j)}^{-2}$ converges, so $w$ is a sequence of Wick couplings relative to this pair, with bound $|w_{0}|$.

3. (The Riccati condition)¶ Let $\lambda_{0},\theta\in\mathbb{R}$ be positive with $\varepsilon_{0}=1+\lambda_{0}+2\theta\min\{w_{0},0\}>0$. Then

$$\frac{a_{j}}{c_{j}}+\lambda_{0}+2\theta\,w_{j}c_{j}\ge\varepsilon_{0}\qquad\text{for every }j\in\mathbb{N}.$$

4. (The Wick constant diverges)¶ Suppose $2\le n$ and $0<w_{0}$. Then the series $\sum_{j=1}^{\infty}w_{j}c_{j}=\sum_{j=1}^{\infty}w_{0}\,\mu_{\kappa(j)}^{-1}$ does not converge.