The Score-Paired Wick-Square Cost is the Wick-Ordered Series, and Cutoff Counterterms Are Forced up to a Convergent Constant

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, so that the reference measure is $\rho=\gamma_{c}$, let $\beta,\kappa\in\mathbb{R}$ be positive with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian entropy pair with temperature $\beta$, whose hypothesis holds with this $\kappa$, and let $w$ be a sequence of Wick couplings, with score-paired Wick-square cost $G_{w}:\mathcal{D}_{\Sigma}\to\mathbb{R}$. The set $\mathcal{D}_{\Sigma}$ is nonempty, by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty, the pair being a noise penalty pair by The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair. The coordinates $x_{k}$ are those of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §background. For $\nu\in\mathcal{D}_{\Sigma}$ and $k\in\mathbb{N}$ the number $\int_{X}x_{k}^{2}\,\nu(dx)$ is real, since $\nu\in\mathcal{P}_{2}(X)$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain. Convergent series and limits of real sequences are those of those definitions.

1. (The Wick-ordered series)¶ For every $\nu\in\mathcal{D}_{\Sigma}$ the series $\sum_{k=1}^{\infty}w_{k}\bigl(\int_{X}x_{k}^{2}\,\nu(dx)-c_{k}\bigr)$ converges, and its sum is $G_{w}(\nu)$.

2. (Counterterms are forced)¶ Let $(C_{N})_{N\in\mathbb{N}}$ be a real sequence, and for $\nu\in\mathcal{D}_{\Sigma}$ and $N\in\mathbb{N}$ let

$$G_{N}(\nu)=\sum_{k=1}^{N}w_{k}\int_{X}x_{k}^{2}\,\nu(dx)-C_{N},\qquad e_{N}=\sum_{k=1}^{N}w_{k}c_{k}-C_{N}.$$

Then $\lim_{N\to\infty}\bigl(G_{N}(\nu)-e_{N}\bigr)=G_{w}(\nu)$ for every $\nu\in\mathcal{D}_{\Sigma}$. Consequently the following are equivalent: (i) the sequence $(G_{N}(\nu))_{N\in\mathbb{N}}$ converges for some $\nu\in\mathcal{D}_{\Sigma}$; (ii) it converges for every $\nu\in\mathcal{D}_{\Sigma}$; (iii) the sequence $(e_{N})_{N\in\mathbb{N}}$ converges. In that case, with $e_{\infty}$ the limit of $(e_{N})_{N\in\mathbb{N}}$, $\lim_{N\to\infty}G_{N}(\nu)=G_{w}(\nu)+e_{\infty}$ for every $\nu\in\mathcal{D}_{\Sigma}$.

3. (The bare cutoff costs diverge)¶ Suppose that $w_{k}\ge0$ for every $k\in\mathbb{N}$ and that the series $\sum_{k=1}^{\infty}w_{k}c_{k}$ does not converge. Then for every $\nu\in\mathcal{D}_{\Sigma}$ and every $M\in\mathbb{R}$ there is $N_{0}\in\mathbb{N}$ with $\sum_{k=1}^{N}w_{k}\int_{X}x_{k}^{2}\,\nu(dx)>M$ for every $N\in\mathbb{N}$ with $N\ge N_{0}$.