The $Phi^4$ Measure on the Torus of Dimension at Most Two as the Gibbs Measure of a Gibbs-Sobolev Potential, and Its Galerkin Approximation in Total Variation

In the settings of The Real Numbers: Standing Notation and Background, The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation and Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, the last read with the space $X=H^{-m}(\mathbb{T}^{n})$, the orthonormal basis $(e_{j})_{j\in\mathbb{N}}$, the noise weights $a$ and the variance sequence $c$ of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §gaussian, so that $\gamma_{c}$ is the free field, suppose $n\le2$, and let $\varkappa,\mathfrak{m},\beta\in\mathbb{R}$ with $0<\varkappa$ and $0<\beta$. Let $V_{N}$ ($N\in\mathbb{N}$) be the Galerkin Wick-ordered $\phi^{4}$ potentials and $V$ the Wick-ordered $\phi^{4}$ potential with coupling $\varkappa$ and mass $\mathfrak{m}$; $\exp$ is the exponential function and $|s|$ the absolute value of $s\in\mathbb{R}$.

1. (A Gibbs-Sobolev potential)¶ For every real $q>1$, $V$ is a Gibbs-Sobolev potential at temperature $\beta$ with exponent $q$.

2. (Galerkin approximation)¶ The functions $V$ and $V_{N}$ ($N\in\mathbb{N}$) are Borel, and $\exp(-V/\beta)$ and $\exp(-V_{N}/\beta)$ are integrable with respect to $\gamma_{c}$, so their Gibbs measures $\gamma^{V}_{\beta}$ and $\gamma^{V_{N}}_{\beta}$ at temperature $\beta$, with normalisers $Z_{V,\beta}$ and $Z_{V_{N},\beta}$, are defined. The sequence $(Z_{V_{N},\beta})_{N\in\mathbb{N}}$ converges to $Z_{V,\beta}$, and for every positive $\varepsilon\in\mathbb{R}$ there is $N_{0}\in\mathbb{N}$ with

$$\bigl|\gamma^{V_{N}}_{\beta}(A)-\gamma^{V}_{\beta}(A)\bigr|\le\varepsilon\qquad\text{for every }N\ge N_{0}\text{ and every }A\in\mathcal{B}(X).$$