Proof of 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

Each result cited is universally quantified over the data in its own statement.

Fix $j\in\mathbb{N}$ and abbreviate $r=\rho_{\kappa(j)}$ and $\mu=\mu_{\kappa(j)}$. By Summability of the Negative Powers of the Fourier Weights of the Torus §product (used with its $s$ taken to be $n$), $1\le\mu$; hence $\mu$ is positive and $0<\mu^{-1}\le1$. The enumeration $\kappa$ of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §data is a bijection $\mathbb{N}\to\mathbb{Z}^{n}$, by the meaning of an enumeration in Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families. 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, $a_{j}=\mu\,c_{j}$, and $c_{j}>0$ by Variance Sequences and Their Truncations §variances, $c$ being a variance sequence 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 §variances; hence

$$c_{j}=\mu^{-1}a_{j},\qquad\frac{a_{j}}{c_{j}}=\mu,\qquad\frac{c_{j}}{a_{j}}=\mu^{-1}.\tag{1}$$

Since $r^{2}=\mu^{-1}$, the product rule for natural powers (Properties of Natural Number Powers in a Field §products) and its unit rule (Properties of Natural Number Powers in a Field §unit) give

$$a_{j}=(r^{m})^{2}=r^{m}r^{m}=(r\,r)^{m}=(\mu^{-1})^{m},\qquad\mu^{m}a_{j}=(\mu\,\mu^{-1})^{m}=1^{m}=1.\tag{2}$$

Clause 1. Let $x\in X$. The coordinate $x_{j}$ is $\langle x,e_{j}\rangle_{H^{-m}}$ (A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §background with The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §gaussian), which equals $r^{m}\,x(\kappa(j))$ 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 §basis. Hence, using (2),

$$w_{j}\,x_{j}^{2}=w_{0}\,\mu^{m}\,(r^{m})^{2}\,x(\kappa(j))^{2}=w_{0}\,\mu^{m}a_{j}\,x(\kappa(j))^{2}=w_{0}\,x(\kappa(j))^{2}.$$

By (1) and (2), $w_{j}c_{j}=w_{0}\,\mu^{m}\mu^{-1}a_{j}=w_{0}\,\mu^{-1}$. Since $c_{j}>0$ and $0<\mu^{-1}\le1$,

$$|w_{j}|\,c_{j}=|w_{j}c_{j}|=|w_{0}|\,\mu^{-1}\le|w_{0}| .$$

Clause 2. Suppose $n\le3$. By clause 1 and (1),

$$\frac{|w_{j}|\,c_{j}^{2}}{a_{j}}=|w_{j}|\,c_{j}\cdot\frac{c_{j}}{a_{j}}=|w_{0}|\,\mu^{-1}\mu^{-1}=|w_{0}|\,\mu^{-2},$$

$\mu^{-1}\mu^{-1}$ being the inverse of $\mu^{2}=\mu\mu$. Since $n\le3<4=2\cdot2$, Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §convergent with $s=2$ shows that the family $k\mapsto1/\mu_{k}^{2}$ on $\mathbb{Z}^{n}$ is cube-summable. Its values are positive, so by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §enumeration, applied with the bijection $\kappa$, the series $\sum_{j=1}^{\infty}\mu_{\kappa(j)}^{-2}$ converges; by Elementary Properties of Series of Real Numbers §linearity so does $\sum_{j=1}^{\infty}|w_{0}|\,\mu_{\kappa(j)}^{-2}$, which is, term by term, the series $\sum_{j=1}^{\infty}|w_{j}|\,c_{j}^{2}/a_{j}$. Together with $|w_{j}|\,c_{j}\le|w_{0}|$ for every $j$ (clause 1), this is the condition of The Wick-Square Corrector and the Score-Paired Wick-Square Cost Relative to a Diagonal Gaussian Measure on a Hilbert Space §couplings, so $w$ is a sequence of Wick couplings with bound $|w_{0}|$. That definition applies to the pair of the statement: $\beta=1$ and the constant $1$ are positive, and $c_{j}\le1\cdot a_{j}$ for every $j$ 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.

Clause 3. Let $\lambda_{0},\theta$ be positive with $\varepsilon_{0}=1+\lambda_{0}+2\theta\min\{w_{0},0\}>0$. By (1) and clause 1, with $t=\mu\ge1$,

$$\frac{a_{j}}{c_{j}}+\lambda_{0}+2\theta\,w_{j}c_{j}=t+\lambda_{0}+\frac{2\theta w_{0}}{t}.$$

Suppose first $0\le w_{0}$. Then $\min\{w_{0},0\}=0$ by Minimum of Two Elements of a Totally Ordered Set: it is $w_{0}$ if $w_{0}\le0$, in which case $w_{0}=0$, and $0$ otherwise, so $\varepsilon_{0}=1+\lambda_{0}$; since $t\ge1$ and $2\theta w_{0}/t\ge0$, the displayed quantity is at least $1+\lambda_{0}=\varepsilon_{0}$. Suppose next $w_{0}<0$. Then $\min\{w_{0},0\}=w_{0}$ by Minimum of Two Elements of a Totally Ordered Set, so $\varepsilon_{0}=1+\lambda_{0}+2\theta w_{0}$, and

$$t+\lambda_{0}+\frac{2\theta w_{0}}{t}-\varepsilon_{0}=(t-1)+2\theta w_{0}\Bigl(\frac{1}{t}-1\Bigr)=(t-1)-\frac{2\theta w_{0}(t-1)}{t}=(t-1)\Bigl(1-\frac{2\theta w_{0}}{t}\Bigr).$$

Here $t-1\ge0$, and $-2\theta w_{0}/t>0$ because $\theta,t>0$ and $w_{0}<0$, so the second factor exceeds $1$; the product is nonnegative. In both cases $\frac{a_{j}}{c_{j}}+\lambda_{0}+2\theta w_{j}c_{j}\ge\varepsilon_{0}$, and $j$ was arbitrary.

Clause 4. Suppose $2\le n$ and $0<w_{0}$. By clause 1, $w_{j}c_{j}=w_{0}\,\mu_{\kappa(j)}^{-1}$ for every $j$, so the two series of the claim agree term by term. The family $k\mapsto1/\mu_{k}$ on $\mathbb{Z}^{n}$ has positive values, and by Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §divergent the set of its cube sums is not bounded above, so it is not cube-summable by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative. By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §enumeration, applied with the bijection $\kappa$, the series $\sum_{j=1}^{\infty}\mu_{\kappa(j)}^{-1}$ does not converge. If $\sum_{j=1}^{\infty}w_{0}\,\mu_{\kappa(j)}^{-1}$ converged, then multiplying by $w_{0}^{-1}$, which exists as $w_{0}>0$, Elementary Properties of Series of Real Numbers §linearity would make $\sum_{j=1}^{\infty}\mu_{\kappa(j)}^{-1}$ converge, a contradiction. Hence $\sum_{j=1}^{\infty}w_{j}c_{j}=\sum_{j=1}^{\infty}w_{0}\,\mu_{\kappa(j)}^{-1}$ does not converge.