The Hubbard-Stratonovich Field Law is the Gibbs Measure of an Admissible Cylindrical Potential for Every Admissible Split of Its Gaussian, and the Gibbs Entropy Pair Does Not Depend on the Split

We use clauses The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §torus and The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §data of the free-field setting, but not its clause on the Gaussian reference: $X=H^{-m}(\mathbb{T}^{n})$ with orthonormal basis $(e_{j})_{j\in\mathbb{N}}$, the noise weights $a_{j}$, which form a weight sequence bounded by $1$, the free-field variances $c_{j}$, the enumeration $\kappa$ and the values $x(k)$. Let $M\in\mathbb{N}$, let $\mathfrak{j}$ be a reflection-symmetric periodic pair interaction of side $L=2M+1$ with symbol $\hat{J}$, let $\eta\in\mathbb{R}$ be a shift for $\mathfrak{j}$, with $\mathfrak{q}_{k}=\hat{J}(k)+\eta$ positive for $k$ in the cube $\Gamma_{M}$, and let $\gamma^{\mathrm{HS}}_{J,\eta}$ be the Hubbard-Stratonovich field law. The lattice torus $\mathbb{L}_{M}$, the maps $\psi_{k}$, $L^{\pm n/2}$ and $\exp$ are as in The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus, and $\operatorname{lch}$ is the function of The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus §field-to-spin; $\mathcal{E}_{M}$ is the white-noise embedding and $\varphi_{x}$ the site field of $x\in X$; $d_{M}$, $\chi_{M}$, $a_{j}^{\pm1/2}$ and the coordinates $x_{j}$ are as in The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates; $\log$ is the natural logarithm; and finite relative entropy and $H(\cdot\,|\,\cdot)$ are those of Relative Entropy of Probability Measures §relative-entropy. An admissible split is a real number $\varsigma$ with $\mathfrak{q}_{k}<\varsigma$ for every $k\in\Gamma_{M}$; it is positive, since $\Gamma_{M}$ is nonempty by The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite and each $\mathfrak{q}_{k}$ is positive. For an admissible split $\varsigma$, claims 2 to 5 are read in the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation with the space $X$, the basis $(e_{j})$, the noise weights $a$ with bound $1$, and the variance sequence $c^{(\varsigma)}$ of claim 1 in place of $c$, and with $N$ in place of the natural number written $n$ there, as in The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §gaussian, since $n$ is the dimension of the torus; so there the reference measure is $\rho=\gamma_{c^{(\varsigma)}}$, the diagonal Gaussian measure on $X$ with variances $c^{(\varsigma)}$, and $X^{a}$, $|\cdot|_{a}$, $p_{N}$ and $\mathcal{P}^{a}_{\rho}$ are as there. The letter $\rho$ denotes $\gamma_{c^{(\varsigma)}}$ only in this reading.

1. (The split Gaussian)¶ Let $\varsigma$ be an admissible split. For $k\in\Gamma_{M}$ put $r^{(\varsigma)}_{k}=\mathfrak{q}_{k}\varsigma/(\varsigma-\mathfrak{q}_{k})$, a positive number with $1/r^{(\varsigma)}_{k}=1/\mathfrak{q}_{k}-1/\varsigma$; for $j\in\mathbb{N}$ put $c^{(\varsigma)}_{j}=a_{j}\,r^{(\varsigma)}_{\kappa(j)}$ if $\kappa(j)\in\Gamma_{M}$ and $c^{(\varsigma)}_{j}=c_{j}$ otherwise. Then $c^{(\varsigma)}$ is a variance sequence, and $c^{(\varsigma)}_{j}\le\bar{r}_{\varsigma}\,a_{j}$ for every $j\in\mathbb{N}$, where $\bar{r}_{\varsigma}$ is the greatest of $1$ and the finitely many numbers $r^{(\varsigma)}_{k}$ ($k\in\Gamma_{M}$), which exists by Greatest Element of a Finite Family in a Totally Ordered Set, applied to the tuple consisting of $1$ followed by these numbers along an enumeration of the finite set $\Gamma_{M}$.

2. (The potential)¶ Let $f_{\varsigma}:\mathbb{R}\to\mathbb{R}$, $f_{\varsigma}(t)=\frac{t^{2}}{2\varsigma}-\operatorname{lch}(t)$, and let $V_{\varsigma}:X\to\mathbb{R}$ and $v_{\varsigma}:\mathbb{R}^{d_{M}}\to\mathbb{R}$ be the maps

$$V_{\varsigma}(x)=\sum_{z\in\mathbb{L}_{M}}f_{\varsigma}\bigl(\varphi_{x}(z)\bigr),\qquad v_{\varsigma}(u)=\sum_{z\in\mathbb{L}_{M}}f_{\varsigma}\Bigl(L^{-n/2}\sum_{j=1}^{d_{M}}\chi_{M}(j)\,a_{j}^{-1/2}u_{j}\,\psi_{\kappa(j)}(z)\Bigr).$$

Then $V_{\varsigma}=v_{\varsigma}\circ p_{d_{M}}$ by The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates §coordinates, and $V_{\varsigma}$ is an admissible cylindrical potential with head dimension $d_{M}$, profile $v_{\varsigma}$ and semiconvexity constant $K_{\varsigma}$, the greater of $0$ and $1-\varsigma^{-1}$; the constant $b=L^{n}\bigl(\frac{\varsigma}{2}+\log2\bigr)$, with $L^{n}$ the natural power, serves in its lower bound.

3. (The noise gradient)¶ The noise gradient of $V_{\varsigma}$ is $\nabla_{a}V_{\varsigma}(x)=\mathcal{E}_{M}\bigl(f_{\varsigma}'\circ\varphi_{x}\bigr)$ for $x\in X$, where $f_{\varsigma}'(t)=\frac{t}{\varsigma}-\frac{\exp(t)-\exp(-t)}{\exp(t)+\exp(-t)}$ is the derivative of $f_{\varsigma}$.

4. (The field law is a Gibbs measure)¶ The Gibbs measure $\gamma^{V_{\varsigma}}_{1}$ of $V_{\varsigma}$ at temperature $1$ relative to $\gamma_{c^{(\varsigma)}}$ equals $\gamma^{\mathrm{HS}}_{J,\eta}$.

5. (Independence of the split)¶ Let $\varsigma$ and $\varsigma'$ be admissible splits. Then $\mathcal{P}^{a}_{\gamma_{c^{(\varsigma)}}}=\mathcal{P}^{a}_{\gamma_{c^{(\varsigma')}}}$, and the Gibbs entropy pair with potential $V_{\varsigma}$ and temperature $1$ relative to $\gamma_{c^{(\varsigma)}}$, whose hypothesis holds with the constant $\bar{r}_{\varsigma}$ by claim 1, is the same quadruple $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ as the one with $\varsigma'$ in place of $\varsigma$. Its penalty domain $\mathcal{D}$ is the set of the $\mu\in\mathcal{P}(X)$ of finite relative entropy with respect to $\gamma^{\mathrm{HS}}_{J,\eta}$, and $\mathcal{E}(\mu)=H(\mu\,|\,\gamma^{\mathrm{HS}}_{J,\eta})$ for $\mu\in\mathcal{D}$.