The Hubbard-Stratonovich Transform Couples the Ising Measure and the Field Law: the Density of the Field Law, the Disintegration of the Joint Law, and Entropy Contraction under Both Kernels

In the setting of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation, let $M\in\mathbb{N}$, and let the side $L=2M+1$ and the cube $\Gamma_{M}$ be those of The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field. Let $\mathfrak{j}$ be a reflection-symmetric periodic pair interaction of side $L$ with interaction matrix $J$ and symbol $\hat{J}$, and let $\eta\in\mathbb{R}$ be a shift for $\mathfrak{j}$, so that $\mathfrak{q}_{k}=\hat{J}(k)+\eta>0$ for every $k\in\Gamma_{M}$. Let $\gamma_{c^{J,\eta}}$ be the Gaussian reference, $K^{\uparrow}_{J,\eta}$ the spin-to-field kernel, $K^{\downarrow}$ the field-to-spin kernel, $\lambda K^{\uparrow}_{J,\eta}$ and $\nu K^{\downarrow}$ the transported laws, $\gamma^{\mathrm{HS}}_{J,\eta}$ the field law and $\Xi_{J,\eta}$ the joint law of The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus, with the lattice torus $\mathbb{L}_{M}$, the site fields $\varphi_{x}$, $\mathcal{B}(X)$, $\mathcal{P}(X)$, $\exp$ and $\operatorname{lch}$ as there. Let $\mathcal{C}_{M}$ be the set of spin configurations, with power set $2^{\mathcal{C}_{M}}$, let $\mathbb{P}_{J}$ be the Ising measure of $\mathfrak{j}$, and let $Z_{J}$ be its partition function. Integrals and integrability are those of Measure Spaces and the Lebesgue Integral: Standing Notation, with $\int h(t)\,\lambda(dt)$ and indicators $\mathbf{1}_{A}$ as in Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral. Finite relative entropy and $H(\cdot\,|\,\cdot)$ are those of Relative Entropy of Probability Measures §relative-entropy, on $(X,\mathcal{B}(X))$ and on $(\mathcal{C}_{M},2^{\mathcal{C}_{M}})$, and $L^{n}$ is the natural power of $L$ in $\mathbb{R}$.

1. (Density of the field law)¶ Let $\varrho^{\mathrm{HS}}_{J,\eta}:X\to\mathbb{R}$ be the map

$$\varrho^{\mathrm{HS}}_{J,\eta}(x)=\frac{1}{Z_{J}}\exp\Bigl(-\frac{\eta L^{n}}{2}+\sum_{z\in\mathbb{L}_{M}}\operatorname{lch}\bigl(\varphi_{x}(z)\bigr)\Bigr).$$

It is positive and continuous with respect to the distance of $X$, $\int_{X}\varrho^{\mathrm{HS}}_{J,\eta}\,d\gamma_{c^{J,\eta}}=1$, and

$$\gamma^{\mathrm{HS}}_{J,\eta}(B)=\int_{X}\mathbf{1}_{B}\,\varrho^{\mathrm{HS}}_{J,\eta}\,d\gamma_{c^{J,\eta}}\qquad\text{for every }B\in\mathcal{B}(X).$$

2. (Disintegration of the joint law)¶ For every $A\subseteq\mathcal{C}_{M}$ and every $B\in\mathcal{B}(X)$,

$$\Xi_{J,\eta}(A\times B)=\int_{X}\mathbf{1}_{B}(x)\,K^{\downarrow}(x,A)\,\gamma^{\mathrm{HS}}_{J,\eta}(dx).$$

3. (Marginals)¶ $\gamma^{\mathrm{HS}}_{J,\eta}K^{\downarrow}=\mathbb{P}_{J}$.

4. (Entropy contraction under the field-to-spin kernel)¶ Let $\nu\in\mathcal{P}(X)$ have finite relative entropy with respect to $\gamma^{\mathrm{HS}}_{J,\eta}$. Then $\nu K^{\downarrow}$ has finite relative entropy with respect to $\mathbb{P}_{J}$, and

$$H(\nu K^{\downarrow}\,|\,\mathbb{P}_{J})\le H(\nu\,|\,\gamma^{\mathrm{HS}}_{J,\eta}).$$

5. (Entropy contraction under the spin-to-field kernel)¶ Let $\lambda$ be a probability measure on $(\mathcal{C}_{M},2^{\mathcal{C}_{M}})$. Then $\lambda$ has finite relative entropy with respect to $\mathbb{P}_{J}$, $\lambda K^{\uparrow}_{J,\eta}$ has finite relative entropy with respect to $\gamma^{\mathrm{HS}}_{J,\eta}$, and

$$H(\lambda K^{\uparrow}_{J,\eta}\,|\,\gamma^{\mathrm{HS}}_{J,\eta})\le H(\lambda\,|\,\mathbb{P}_{J}).$$