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Now if we have a complex-valued test function $f\in\mathscr{S}(\mathbb{R},\mathbb{C})$, it holds that $$ \int_{\mathscr{S}'(\mathbb{R}^d,\mathbb{R})} e^{i\varphi(f)}\ d\mu_M(\varphi) =\left\langle \exp\left[i\sum_{x\in\mathbb{Z}}\sigma_x g_x\right] \right\rangle $$ where $g=(g_x)_{x\in\mathbb{Z}}\in \mathcal{s}(\mathbb{Z},\mathbb{C})$ is a discrete test function, with rapid decay on the lattice $\mathbb{Z}$, given by $g_x=L^{-\frac{M}{2}}f(L^{-M}x)$. Since we work with complex-valued test functions, the difference between characteristic functions and moment generating function is moot and amounts to deciding to absorb the $i$ into the test function or not. We will keep it out.

With the above result one can show that for $f\in\mathscr{S}(\mathbb{R},\mathbb{C})$ such that $4eK||f||_{L^{\infty}}||f||_{L^{1}}<1$, $$ \lim\limits_{M\rightarrow \infty} \int_{\mathscr{S}'(\mathbb{R}^d,\mathbb{R})} e^{i\varphi(f)}\ d\mu_M(\varphi) = \exp\left[-\frac{K}{2}\int_{\mathbb{R}}f(x)^2\ dx\right]\ . $$ Here the constant $K$ is related to the susceptibility and is given by $K=\frac{1+e^{-m}}{1-e^{-m}}$.

Now take $f\in\mathscr{S}(\mathbb{R},\mathbb{R})$, a real-valued test function but this time with no limitation on its size. For $z\in\mathbb{C}$, let $$ G_M(z)=\int_{\mathscr{S}'(\mathbb{R}^d,\mathbb{R})} e^{iz\varphi(f)}\ d\mu_M(\varphi)\ , $$ which is well defined and analytic in an open horizontal strip around the real axis. The bound for $\mathcal{F}$ shows that $G_M(z)$ is uniformly bounded in $z$ and $M$ on every compact subset of the strip. The Vitali-Porter Theorem then gives pointwise convergence on the real line, and the Lévy Continuity Theorem finishes the job of proving that we weakly converge to $\sqrt{K}$ times white noise.

Now if we have a complex-valued test function $f\in\mathscr{S}(\mathbb{R},\mathbb{C})$, it holds that $$ \int_{\mathscr{S}'(\mathbb{R},\mathbb{R})} e^{i\varphi(f)}\ d\mu_M(\varphi) =\left\langle \exp\left[i\sum_{x\in\mathbb{Z}}\sigma_x g_x\right] \right\rangle $$ where $g=(g_x)_{x\in\mathbb{Z}}\in \mathcal{s}(\mathbb{Z},\mathbb{C})$ is a discrete test function, with rapid decay on the lattice $\mathbb{Z}$, given by $g_x=L^{-\frac{M}{2}}f(L^{-M}x)$. Since we work with complex-valued test functions, the difference between characteristic functions and moment generating function is moot and amounts to deciding to absorb the $i$ into the test function or not. We will keep it out.

With the above result one can show that for $f\in\mathscr{S}(\mathbb{R},\mathbb{C})$ such that $4eK||f||_{L^{\infty}}||f||_{L^{1}}<1$, $$ \lim\limits_{M\rightarrow \infty} \int_{\mathscr{S}'(\mathbb{R},\mathbb{R})} e^{i\varphi(f)}\ d\mu_M(\varphi) = \exp\left[-\frac{K}{2}\int_{\mathbb{R}}f(x)^2\ dx\right]\ . $$ Here the constant $K$ is related to the susceptibility and is given by $K=\frac{1+e^{-m}}{1-e^{-m}}$.

Now take $f\in\mathscr{S}(\mathbb{R},\mathbb{R})$, a real-valued test function but this time with no limitation on its size. For $z\in\mathbb{C}$, let $$ G_M(z)=\int_{\mathscr{S}'(\mathbb{R},\mathbb{R})} e^{iz\varphi(f)}\ d\mu_M(\varphi)\ , $$ which is well defined and analytic in an open horizontal strip around the real axis. The bound for $\mathcal{F}$ shows that $G_M(z)$ is uniformly bounded in $z$ and $M$ on every compact subset of the strip. The Vitali-Porter Theorem then gives pointwise convergence on the real line, and the Lévy Continuity Theorem finishes the job of proving that we weakly converge to $\sqrt{K}$ times white noise.

Consider the Ising model in one dimension with zero magnetic field, nearest-neighbor coupling function $J>0$ and inverse temperature $\beta$, in infinite volume. This is a Borel probability measure $\mu$ on $\{-1,1\}^{\mathbb{Z}}$, with corresponding expectations denoted by $\langle \cdots\rangle$. Pick, once and for all, your favorite number $L>1$ and for $M\in\mathbb{N}$, define the map $$ \begin{array}{cccc} \Gamma_M: & \{-1,1\}^{\mathbb{Z}} & \longrightarrow & \mathscr{S}'(\mathbb{R},\mathbb{R})\ \\ & \sigma=(\sigma_x)_{x\in\mathbb{Z}} & \longmapsto & L^{-\frac{M}{2}}\sum\limits_{x\in\mathbb{Z}}\sigma_x \delta_{L^{-M}x} \end{array} $$ where $\delta_z$ denotes the Dirac distribution located at $z$. This map is well defined, continuous, and therefore Borel measurable. This allows you to define the push-forward measure $\mu_M:=(\Gamma_M)_{\ast}\mu$. It turns out that when $M\rightarrow\infty$ the measure $\mu_M$ converges weakly to a multiple of white noise on $\mathbb{R}$. This is a special case of a theorem of Newman for FKG spin systems, but it is a good exercise to do it using, in a completely explicit way, characteristic functions/moment generating functions via the control of what is happening in a complex neighborhood of the origin.

Consider the Ising model in one dimension with zero magnetic field, nearest-neighbor coupling function $J>0$ and inverse temperature $\beta$, in infinite volume. This is a Borel probability measure $\mu$ on $\{-1,1\}^{\mathbb{Z}}$, with corresponding expectations denoted by $\langle \cdots\rangle$. Pick, once and for all, your favorite number $L>1$ and for $M\in\mathbb{N}$, define the map $$ \begin{array}{cccc} \Gamma_M: & \{-1,1\}^{\mathbb{Z}} & \longrightarrow & \mathscr{S}'(\mathbb{R},\mathbb{R})\ \\ & \sigma=(\sigma_x)_{x\in\mathbb{Z}} & \longmapsto & L^{-\frac{M}{2}}\sum\limits_{x\in\mathbb{Z}}\sigma_x \delta_{L^{-M}x} \end{array} $$ where $\delta_z$ denotes the Dirac distribution located at $z$. Similarly to my answer to

A set of questions on continuous Gaussian Free Fields (GFF)

this map is well defined, continuous, and therefore Borel measurable. This allows you to define the push-forward measure $\mu_M:=(\Gamma_M)_{\ast}\mu$. It turns out that when $M\rightarrow\infty$ the measure $\mu_M$ converges weakly to a multiple of white noise on $\mathbb{R}$. This is a special case of a theorem of Newman for FKG spin systems, but it is a good exercise to do it using, in a completely explicit way, characteristic functions/moment generating functions via the control of what is happening in a complex neighborhood of the origin.

Now if we have a complex-valued test function $f\in\mathscr{S}(\mathbb{R},\mathbb{C})$, it holds that $$ \mathbb{E}_{\mu_M}[e^{i\phi(f)}]=\left\langle \exp\left[i\sum_{x\in\mathbb{Z}}\sigma_x g_x\right] \right\rangle $$ where $g=(g_x)_{x\in\mathbb{Z}}\in \mathcal{s}(\mathbb{Z},\mathbb{C})$ is a discrete test function, with rapid decay on the lattice $\mathbb{Z}$, given by $g_x=L^{-\frac{M}{2}}f(L^{-M}x)$. Since we work with complex-valued test functions, the difference between characteristic functions and moment generating function is moot and amounts to deciding to absorb the $i$ into the test function or not. We will keep it out.

With the above result one can show that for $f\in\mathscr{S}(\mathbb{R},\mathbb{C})$ such that $4eK||f||_{L^{\infty}}||f||_{L^{1}}<1$, $$ \lim\limits_{M\rightarrow \infty} \mathbb{E}_{\mu_M}[e^{i\phi(f)}]= \exp\left[-\frac{K}{2}\int_{\mathbb{R}}f(x)^2\ dx\right]\ . $$ Here the constant $K$ is related to the susceptibility and is given by $K=\frac{1+e^{-m}}{1-e^{-m}}$.

Now take $f\in\mathscr{S}(\mathbb{R},\mathbb{R})$, a real-valued test function but this time with no limitation on its size. For $z\in\mathbb{C}$, let $$ G_M(z)=\mathbb{E}_{\mu_M}[e^{iz\phi(f)}]\ , $$ which is well defined and analytic in an open horizontal strip around the real axis. The bound for $\mathcal{F}$ shows that $G_M(z)$ is uniformly bounded in $z$ and $M$ on every compact subset of the strip. The Vitali-Porter then gives pointwise convergence on the real line, and the Lévy Continuity Theorem finishes the job of proving that we weakly converge to $\sqrt{K}$ times white noise.

Now if we have a complex-valued test function $f\in\mathscr{S}(\mathbb{R},\mathbb{C})$, it holds that $$ \int_{\mathscr{S}'(\mathbb{R}^d,\mathbb{R})} e^{i\varphi(f)}\ d\mu_M(\varphi) =\left\langle \exp\left[i\sum_{x\in\mathbb{Z}}\sigma_x g_x\right] \right\rangle $$ where $g=(g_x)_{x\in\mathbb{Z}}\in \mathcal{s}(\mathbb{Z},\mathbb{C})$ is a discrete test function, with rapid decay on the lattice $\mathbb{Z}$, given by $g_x=L^{-\frac{M}{2}}f(L^{-M}x)$. Since we work with complex-valued test functions, the difference between characteristic functions and moment generating function is moot and amounts to deciding to absorb the $i$ into the test function or not. We will keep it out.

With the above result one can show that for $f\in\mathscr{S}(\mathbb{R},\mathbb{C})$ such that $4eK||f||_{L^{\infty}}||f||_{L^{1}}<1$, $$ \lim\limits_{M\rightarrow \infty} \int_{\mathscr{S}'(\mathbb{R}^d,\mathbb{R})} e^{i\varphi(f)}\ d\mu_M(\varphi) = \exp\left[-\frac{K}{2}\int_{\mathbb{R}}f(x)^2\ dx\right]\ . $$ Here the constant $K$ is related to the susceptibility and is given by $K=\frac{1+e^{-m}}{1-e^{-m}}$.

Now take $f\in\mathscr{S}(\mathbb{R},\mathbb{R})$, a real-valued test function but this time with no limitation on its size. For $z\in\mathbb{C}$, let $$ G_M(z)=\int_{\mathscr{S}'(\mathbb{R}^d,\mathbb{R})} e^{iz\varphi(f)}\ d\mu_M(\varphi)\ , $$ which is well defined and analytic in an open horizontal strip around the real axis. The bound for $\mathcal{F}$ shows that $G_M(z)$ is uniformly bounded in $z$ and $M$ on every compact subset of the strip. The Vitali-Porter Theorem then gives pointwise convergence on the real line, and the Lévy Continuity Theorem finishes the job of proving that we weakly converge to $\sqrt{K}$ times white noise.