The Wick-Ordered $Phi^4$ Potential of the Free Field on the Torus of Dimension at Most Two

In the settings of The Real Numbers: Standing Notation and Background and The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation, suppose $n\le2$, let $\varkappa,\mathfrak{m}\in\mathbb{R}$ with $0<\varkappa$, and for $N\in\mathbb{N}$ let $V_{N}:X\to\mathbb{R}$ be the Galerkin Wick-ordered $\phi^{4}$ potential with cutoff $N$, coupling $\varkappa$ and mass $\mathfrak{m}$; $2^{j}$ ($j\in\mathbb{N}$) is the $j$-th power of $2$.

(The potential)¶ The Wick-ordered $\phi^{4}$ potential of the free field on $\mathbb{T}^{n}$ with coupling $\varkappa$ and mass $\mathfrak{m}$ is the function $V:X\to\mathbb{R}$ given, for $x\in X$, by

$$V(x)=\lim_{j\to\infty}V_{2^{j}}(x)\quad\text{if the real sequence }(V_{2^{j}}(x))_{j\in\mathbb{N}}\text{ converges},\qquad V(x)=0\quad\text{otherwise}.$$