Proof of Nelson's Exponential Integrability for the Wick-Ordered $Phi^4$ Potential on the Torus of Dimension at Most Two

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

Let $C\ge0$ be the constant of The Galerkin Wick-Ordered Phi^4 Potentials on the Torus of Dimension at Most Two under the Free Field: Chaos Order, Conditional Expectations, and Increments with Their Cameron-Martin Derivatives §increment, put $C'=C+1>0$, and let $B_{N}$ be the numbers of The Wick-Ordered Phi^4 Potential on the Torus of Dimension at Most Two: Limit of the Galerkin Potentials Almost Everywhere and in Every L^p, Conditional Expectations, Cameron-Martin Derivatives, and the Lower Bound at Each Cutoff §lower-bound, so that $V_{N}\ge-B_{N}$ on $X$. Every $V_{N}$ is Borel by Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient §continuity (applicable by The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass §admissible), and $V$ is Borel by The Wick-Ordered Phi^4 Potential on the Torus of Dimension at Most Two: Limit of the Galerkin Potentials Almost Everywhere and in Every L^p, Conditional Expectations, Cameron-Martin Derivatives, and the Lower Bound at Each Cutoff §borel; for real $s$ the functions $\exp(sV_{N})$ and $\exp(sV)$ are therefore Borel and positive, exactly as in The Gibbs Measure of a Borel Potential Relative to a Diagonal Gaussian Measure §weight. Real powers obey claims 1 and 2 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities, used without further mention; $\mathrm{e}=\exp(1)$.

Step 1 (growth of $B_{N}$). Let $\varepsilon=1/128$ and $S=\sum_{k\in\mathbb{Z}^{n}}\mu_{k}^{-(1+\varepsilon)}\ge0$. For $N\in\mathbb{N}$, $C_{N}=\sum_{k\in\Gamma_{N}}\mu_{k}^{-1}$ by The Galerkin Wick-Ordered Phi^4 Potential on the Torus is an Admissible Cylindrical Potential, with Semiconvexity Constant the Positive Part of Three Times the Coupling Times the Wick Constant Minus the Mass §wick, and since $n\le2$ and $1\le N^{2}$ we have $1+4\pi^{2}nN^{2}\le(1+8\pi^{2})N^{2}$; so Lattice Sums of Powers of the Fourier Weights in Dimension at Most Two: Power Sums, the Growth of the Wick Constants, and the Quartic Convolution Sums §growth gives

$$C_{N}\le\bigl((1+8\pi^{2})N^{2}\bigr)^{1/128}S=A_{1}N^{1/64},\qquad A_{1}=(1+8\pi^{2})^{1/128}S .$$

As $1\le N^{1/64}\le N^{1/32}$, we get $3\varkappa C_{N}+|\mathfrak{m}|\le(3\varkappa A_{1}+|\mathfrak{m}|)N^{1/64}$ and $|\mathfrak{m}|C_{N}\le|\mathfrak{m}|A_{1}N^{1/32}$, hence, squaring the first (both sides nonnegative),

$$B_{N}\le A\,N^{1/32}\quad(N\in\mathbb{N}),\qquad A=\frac{(3\varkappa A_{1}+|\mathfrak{m}|)^{2}}{4\varkappa}+\frac{|\mathfrak{m}|A_{1}}{2}+1>0 .$$

Step 2 (a uniform Gaussian tail). Put $\lambda_{0}=\max\{2A,\,8\mathrm{e}^{2}C'^{1/2},\,1\}$ and $c_{2}=\frac{2}{\mathrm{e}}\,2^{-9/16}C'^{-1/4}(2A)^{-2}>0$. We claim: for every real $\lambda\ge\lambda_{0}$ and every $N\in\mathbb{N}$,

$$\gamma_{c}\bigl(\{x:V_{N}(x)<-\lambda\}\bigr)\le\exp(-c_{2}\lambda^{2}).$$

Fix $\lambda$ and let $y=(\lambda/(2A))^{32}$; then $y\ge1$. By claim 4 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities there is $n_{1}\in\mathbb{N}$ with $y<n_{1}\le y+1$; as $y\ge1$, $n_{1}\ge2$, and $N'=n_{1}-1\in\mathbb{N}$ satisfies $y-1<N'\le y$, hence also $y<2N'$ (add $N'\ge1$). By Step 1, $B_{N'}\le AN'^{1/32}\le Ay^{1/32}=\lambda/2$.

If $N\le N'$, then likewise $B_{N}\le AN^{1/32}\le AN'^{1/32}\le\lambda/2$, so $V_{N}\ge-\lambda/2>-\lambda$ everywhere and the set is empty.

If $N'<N$, let $f=V_{N}-V_{N'}$. Its class lies in $\mathcal{H}_{\le4}$, by The Galerkin Wick-Ordered Phi^4 Potentials on the Torus of Dimension at Most Two under the Free Field: Chaos Order, Conditional Expectations, and Increments with Their Cameron-Martin Derivatives §chaos and since $\mathcal{H}_{\le4}$ is a linear subspace (The Wiener Chaos Decomposition: Cylindrical Hermite Polynomials Are Dense and the Gaussian Lebesgue Space Is the Orthogonal Sum of the Chaoses §subspaces); and by The Galerkin Wick-Ordered Phi^4 Potentials on the Torus of Dimension at Most Two under the Free Field: Chaos Order, Conditional Expectations, and Increments with Their Cameron-Martin Derivatives §increment, $\lVert f\rVert_{2}^{2}\le CN'^{-1/4}\le C'N'^{-1/4}$, so $\lVert f\rVert_{2}\le\sigma:=C'^{1/2}N'^{-1/8}$, and $0<\sigma\le C'^{1/2}$ as $N'\ge1$. If $V_{N}(x)<-\lambda$, then $f(x)<-\lambda+B_{N'}\le-\lambda/2$, so $|f(x)|\ge\lambda/2$. Since $\lambda/2\ge4\mathrm{e}^{2}C'^{1/2}\ge(2\mathrm{e})^{2}\sigma$, Moment and Tail Bounds on the Wiener Chaos up to a Given Order §tail, applied with chaos order $4$ and $s=\lambda/2$, gives

$$\gamma_{c}(\{V_{N}<-\lambda\})\le\gamma_{c}(\{|f|\ge\lambda/2\})\le\exp\Bigl(-\frac{2}{\mathrm{e}}\Bigl(\frac{\lambda}{2\sigma}\Bigr)^{1/2}\Bigr).$$

Now $(\lambda/(2\sigma))^{1/2}=(\lambda/2)^{1/2}C'^{-1/4}N'^{1/16}$, and $N'>y/2$ gives $N'^{1/16}\ge2^{-1/16}y^{1/16}=2^{-1/16}(2A)^{-2}\lambda^{2}$. With $\lambda^{1/2}\ge1$ this yields $\frac{2}{\mathrm{e}}(\lambda/(2\sigma))^{1/2}\ge c_{2}\lambda^{2}$, and the claim follows since $\exp$ is increasing (claim 4 of Basic Properties of the Exponential Function).

Step 3 (claim 1). Fix $t>0$. By The Real Numbers: Standing Notation and Background §background (Archimedean property) choose $R\in\mathbb{N}$ with $R\ge\max\{\lambda_{0},(2t+1)/c_{2}\}$, and put $E_{t}=\exp(tR)+1/(\mathrm{e}-1)$; $E_{t}$ does not depend on $N$. Fix $N$. Let $E_{0}=\{x:V_{N}(x)\ge-R\}$ and, for $k\in\mathbb{N}$, $E_{k}=\{x:-(R+k)\le V_{N}(x)<-(R+k-1)\}$. These are pairwise disjoint Borel sets whose union is $X$ (for $V_{N}(x)<-R$ take the least $k$ with $V_{N}(x)\ge-(R+k)$). The partial sums $h_{K}=\sum_{k=0}^{K}\mathbf{1}_{E_{k}}\exp(-tV_{N})$ increase pointwise to $\exp(-tV_{N})$, so by Monotone Convergence Theorem and Linearity and Monotonicity of the Lebesgue Integral §nonnegative,

$$\int_{X}\exp(-tV_{N})\,d\gamma_{c}=\sup_{K}\sum_{k=0}^{K}\int_{X}\mathbf{1}_{E_{k}}\exp(-tV_{N})\,d\gamma_{c}.$$

On $E_{0}$, $\exp(-tV_{N})\le\exp(tR)$, and $\gamma_{c}(E_{0})\le1$, so the term $k=0$ is at most $\exp(tR)$. For $k\ge1$ put $u=R+k-1\ge R\ge\lambda_{0}$; on $E_{k}$, $\exp(-tV_{N})\le\exp(t(u+1))$, and $E_{k}\subseteq\{V_{N}<-u\}$, so by monotonicity, Step 2 and claim 1 of Basic Properties of the Exponential Function,

$$\int_{X}\mathbf{1}_{E_{k}}\exp(-tV_{N})\,d\gamma_{c}\le\exp(t(u+1))\gamma_{c}(E_{k})\le\exp\bigl(t(u+1)-c_{2}u^{2}\bigr).$$

Since $u\ge1$ and $u\ge(2t+1)/c_{2}$, $c_{2}u^{2}\ge(2t+1)u\ge tu+t+u$, so $t(u+1)-c_{2}u^{2}\le-u\le-k$ and the term is at most $\exp(-k)=\rho^{k}$ with $\rho=\exp(-1)\in(0,1)$ (claims 1, 2 and 4 of Basic Properties of the Exponential Function). By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric, $\sum_{k=1}^{K}\rho^{k}\le\rho/(1-\rho)=1/(\mathrm{e}-1)$. Hence $\int_{X}\exp(-tV_{N})\,d\gamma_{c}\le E_{t}$ for every $N$.

Step 4 (claim 2). Let $f_{i}=\exp(-tV_{2^{i}})$ ($i\in\mathbb{N}$), Borel and nonnegative. By The Wick-Ordered Phi^4 Potential on the Torus of Dimension at Most Two: Limit of the Galerkin Potentials Almost Everywhere and in Every L^p, Conditional Expectations, Cameron-Martin Derivatives, and the Lower Bound at Each Cutoff §limit there is a null set outside which $V_{2^{i}}(x)\to V(x)$, hence $-tV_{2^{i}}(x)\to-tV(x)$ by Arithmetic of Limits of Real Sequences §scalar and $f_{i}(x)\to\exp(-tV(x))$ by claim 3(f) of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities. For a real sequence $a_{i}\to a$ one has $a\le\sup_{k}\inf_{i\ge k}a_{i}$ (given $\varepsilon>0$, $a_{i}>a-\varepsilon$ for all large $i$). So $\exp(-tV)\le\liminf_{i}f_{i}$ almost everywhere, and The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison, Fatou's Lemma and Step 3 give

$$\int_{X}\exp(-tV)\,d\gamma_{c}\le\int_{X}\liminf_{i}f_{i}\,d\gamma_{c}\le\liminf_{i}\int_{X}f_{i}\,d\gamma_{c}\le E_{t},$$

the last because every $\inf_{i\ge k}\int_{X}f_{i}\,d\gamma_{c}$ is at most $E_{t}$. As $\exp(-tV)$ is Borel and positive, it is integrable (Integrable Function and the Lebesgue Integral).

Step 5 (claim 3). Fix real $p\ge1$. For real $u$, $\exp(u)^{p}=\exp(pu)$ by Real Power of a Positive Real Number and The Natural Logarithm; so $\int_{X}|\exp(-tV_{N})|^{p}\,d\gamma_{c}=\int_{X}\exp(-ptV_{N})\,d\gamma_{c}\le E_{pt}$ by Step 3 with $pt$ in place of $t$, and $\int_{X}\exp(-ptV)\,d\gamma_{c}<\infty$ by Step 4 with $pt$. Hence all these classes lie in $L^{p}(\gamma_{c})$, and similarly, with $2p$ in place of $p$, $(\lVert\exp(-tV_{N})\rVert_{2p})^{2p}=\int_{X}\exp(-2ptV_{N})\,d\gamma_{c}\le E_{2pt}$ by Elementary Properties of the p-Seminorm §power, so that $\lVert\exp(-tV_{N})\rVert_{2p}\le E_{2pt}^{1/(2p)}$ by Properties of Real Powers of Nonnegative Real Numbers §monotone and Properties of Real Powers of Nonnegative Real Numbers §inverse, and $D:=\lVert\exp(-tV)\rVert_{2p}<\infty$.

A pointwise bound. For reals $a\ge b$, Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities (claim 6, with $h=a-b\ge0$) and claim 1 of Basic Properties of the Exponential Function give $|\exp(a)-\exp(b)|=\exp(b)|\exp(a-b)-1|\le(a-b)\exp(a)\le|a-b|(\exp(a)+\exp(b))$; by symmetry this holds for all reals $a,b$. With $a=-tV_{N}(x)$ and $b=-tV(x)$,

$$|\exp(-tV_{N})-\exp(-tV)|\le t\,|V_{N}-V|\,\bigl(\exp(-tV_{N})+\exp(-tV)\bigr)\quad\text{on }X.$$

A product bound. If $g,h\in\mathcal{L}^{2p}$, then $gh\in\mathcal{L}^{p}$ and $\lVert gh\rVert_{p}\le\lVert g\rVert_{2p}\lVert h\rVert_{2p}$. Indeed, $|g|^{p},|h|^{p}\in\mathcal{L}^{2}$ with $\lVert\,|g|^{p}\rVert_{2}=\lVert g\rVert_{2p}^{p}$ by Elementary Properties of the p-Seminorm §rescaling; $|gh|^{p}=|g|^{p}|h|^{p}$ by Properties of Real Powers of Nonnegative Real Numbers §product; and Hoelder's Inequality, for Two and for Finitely Many Factors §holder with exponents $2,2$ gives $\int_{X}|gh|^{p}\,d\gamma_{c}\le\lVert g\rVert_{2p}^{p}\lVert h\rVert_{2p}^{p}$; take $p$-th roots by Properties of Real Powers of Nonnegative Real Numbers §monotone and Properties of Real Powers of Nonnegative Real Numbers §exponents.

Conclusion. By The Wick-Ordered Phi^4 Potential on the Torus of Dimension at Most Two: Limit of the Galerkin Potentials Almost Everywhere and in Every L^p, Conditional Expectations, Cameron-Martin Derivatives, and the Lower Bound at Each Cutoff §limit with $2p\ge2$, $V_{N}-V\in\mathcal{L}^{2p}$ and $\lVert V_{N}-V\rVert_{2p}\le(2p-1)^{2}C^{1/2}N^{-1/8}$. By Minkowski's Inequality and the Seminormed Space of Power-Integrable Functions §minkowski, $\lVert\exp(-tV_{N})+\exp(-tV)\rVert_{2p}\le E_{2pt}^{1/(2p)}+D$. The pointwise bound, Elementary Properties of the p-Seminorm §comparison, Elementary Properties of the p-Seminorm §homogeneous and the product bound give

$$\lVert\exp(-tV_{N})-\exp(-tV)\rVert_{p}\le t\,(2p-1)^{2}C^{1/2}\bigl(E_{2pt}^{1/(2p)}+D\bigr)\,N^{-1/8},$$

whose right side tends to $0$ as $N\to\infty$ by claim 3(a) of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities and Arithmetic of Limits of Real Sequences §scalar; so the left side tends to $0$ by claim 3(c) of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities with limit $0$.