The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus

In the setting of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation, with $X=H^{-m}(\mathbb{T}^{n})$, its distance $d$, its orthonormal basis $(e_{j})_{j\in\mathbb{N}}$, the noise weights $a_{j}$, the free-field variances $c_{j}$, the enumeration $\kappa$ of $\mathbb{Z}^{n}$, the Fourier weights $\mu_{k}$ and the values $x(k)$ of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §data and The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §torus, and with the Borel $\sigma$-algebra $\mathcal{B}(X)$, Borel maps, the set $\mathcal{P}(X)$ of Borel probability measures and push-forwards as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures and Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward. Let $M\in\mathbb{N}$; the side $L$, the cube $\Gamma_{M}$, a nonempty finite set by The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite, the maps $\psi_{k}$, the lattice torus $\mathbb{L}_{M}$, lattice fields and their discrete Fourier coefficients are those of The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field; $L^{n/2}$ and the site field $\varphi_{x}$ of $x\in X$ are those of The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution; sums over finite index sets are those of Sum over a Finite Index Set; $\exp$ is the exponential function and $\log$ the natural logarithm. Let $\mathfrak{j}$ be a reflection-symmetric periodic pair interaction of side $L$ with interaction matrix $J$ and symbol $\hat{J}$; the set $\mathcal{C}_{M}$ of spin configurations, its power set $2^{\mathcal{C}_{M}}$ and the Ising measure $\mathbb{P}_{J}$ of $\mathfrak{j}$ are those of The Ising Measure on the Lattice Torus with a Reflection-Symmetric Periodic Pair Interaction, and for a map $p:\mathcal{C}_{M}\to\mathbb{R}$ with $0\le p(\sigma)$ for every $\sigma\in\mathcal{C}_{M}$, $\lambda_{p}$ is the measure with point masses $p$ of Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §measure. Probability kernels are those of Probability Kernels Between Measurable Spaces §kernel, and $\nu\otimes K$ is the composite measure of Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral §composite.

0. (The shift)¶ A real number $\eta$ is a shift for $\mathfrak{j}$ if $\mathfrak{q}_{k}=\hat{J}(k)+\eta>0$ for every $k\in\Gamma_{M}$. In clauses 1 to 6, $\eta$ is a shift for $\mathfrak{j}$ and $\mathfrak{q}_{k}=\hat{J}(k)+\eta$ for $k\in\Gamma_{M}$.

1. (The Gaussian reference)¶ For $j\in\mathbb{N}$ put $c^{J,\eta}_{j}=a_{j}\,\mathfrak{q}_{\kappa(j)}$ if $\kappa(j)\in\Gamma_{M}$ and $c^{J,\eta}_{j}=c_{j}$ otherwise. Then $c^{J,\eta}=(c^{J,\eta}_{j})_{j\in\mathbb{N}}$ is a variance sequence. Its terms are positive, since $a_{j}>0$ by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §weights, a weight sequence having positive terms, $c_{j}>0$ by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §variances, and $\mathfrak{q}_{\kappa(j)}>0$ when $\kappa(j)\in\Gamma_{M}$. Let $D$ be the greatest entry of the tuple consisting of $1$ followed by the numbers $\mu_{k}\mathfrak{q}_{k}$ along an enumeration of $\Gamma_{M}$, which exists by Greatest Element of a Finite Family in a Totally Ordered Set. Since $a_{j}=\mu_{\kappa(j)}c_{j}$ by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §ratio, $c^{J,\eta}_{j}=c_{j}w_{j}$ with $w_{j}=\mu_{\kappa(j)}\mathfrak{q}_{\kappa(j)}$ if $\kappa(j)\in\Gamma_{M}$ and $w_{j}=1$ otherwise, and $0<w_{j}\le D$; so $\sum_{j=1}^{\infty}c^{J,\eta}_{j}$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §tail-bound, applied to the nonnegative terms $c_{j}$, whose series converges by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §variances, and to the factors $w_{j}$ with the bound $D$. The Gaussian reference of the transform is $\gamma_{c^{J,\eta}}$, the diagonal Gaussian measure on $X$ with variances $c^{J,\eta}$.

2. (The spin-to-field kernel)¶ For $\sigma\in\mathcal{C}_{M}$, a lattice field with discrete Fourier coefficients $\hat{\sigma}(k)$, $h_{\sigma}$ is the coefficient family on $\mathbb{Z}^{n}$ with $h_{\sigma}(k)=\mathfrak{q}_{k}L^{n/2}\hat{\sigma}(k)$ for $k\in\Gamma_{M}$ and $h_{\sigma}(k)=0$ otherwise; it belongs to $X$ by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale §series, exactly as in The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution §embedding. For $h\in X$ let $T_{h}:X\to X$, $T_{h}(x)=x+h$; it is continuous, since $d(T_{h}(x),T_{h}(x'))=d(x,x')$, and so Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space. The spin-to-field kernel is

$$K^{\uparrow}_{J,\eta}(\sigma,B)=\bigl((T_{h_{\sigma}})_{\#}\gamma_{c^{J,\eta}}\bigr)(B)\qquad(\sigma\in\mathcal{C}_{M},\ B\in\mathcal{B}(X)).$$

It is a probability kernel from $(\mathcal{C}_{M},2^{\mathcal{C}_{M}})$ to $(X,\mathcal{B}(X))$: each $K^{\uparrow}_{J,\eta}(\sigma,\cdot)$ is the push-forward, as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward, of the Borel probability measure $\gamma_{c^{J,\eta}}$ by the Borel map $T_{h_{\sigma}}$, a Borel measure with total mass $\gamma_{c^{J,\eta}}(T_{h_{\sigma}}^{-1}(X))=\gamma_{c^{J,\eta}}(X)=1$; and for every $B\in\mathcal{B}(X)$ the map $\sigma\mapsto K^{\uparrow}_{J,\eta}(\sigma,B)$ is measurable with respect to $2^{\mathcal{C}_{M}}$ by Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §integral.

3. (The field-to-spin kernel)¶ $\operatorname{lch}:\mathbb{R}\to\mathbb{R}$ is the map $\operatorname{lch}(t)=\log\bigl(\exp(t)+\exp(-t)\bigr)$, defined since $\exp(t)+\exp(-t)>0$ by claim 2 of Basic Properties of the Exponential Function. For $x\in X$ let $\omega_{x}:\mathcal{C}_{M}\to\mathbb{R}$ be the map

$$\omega_{x}(\sigma)=\exp\Bigl(\sum_{z\in\mathbb{L}_{M}}\bigl(\sigma(z)\,\varphi_{x}(z)-\operatorname{lch}(\varphi_{x}(z))\bigr)\Bigr),$$

positive by claim 2 of Basic Properties of the Exponential Function, and let $K^{\downarrow}(x,\cdot)=\lambda_{\omega_{x}}$ be the measure with point masses $\omega_{x}$ on $(\mathcal{C}_{M},2^{\mathcal{C}_{M}})$. The field-to-spin kernel $K^{\downarrow}$ is a probability kernel from $(X,\mathcal{B}(X))$ to $(\mathcal{C}_{M},2^{\mathcal{C}_{M}})$. Indeed, $\sum_{\sigma\in\mathcal{C}_{M}}\omega_{x}(\sigma)=1$ by Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum §spin-sum, applied to the lattice field $\varphi_{x}$, and claims 1 and 2 of Basic Properties of the Exponential Function, so $K^{\downarrow}(x,\cdot)$ is a probability measure by Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §measure. For $A\subseteq\mathcal{C}_{M}$, the map $x\mapsto K^{\downarrow}(x,A)$ is constant $0$ if $A=\emptyset$, and for $A\neq\emptyset$ it is a finite sum of the maps $x\mapsto\omega_{x}(\sigma)$ ($\sigma\in A$). These maps are continuous with respect to the distance $d$ of $X$: each $x\mapsto\varphi_{x}(z)$ is continuous by The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates §site-continuous; $\exp$ is continuous on $\mathbb{R}$ and $\log$ is continuous on $(0,\infty)$ by Differentiability at an Interior Point Implies Continuity There, $\exp$ being differentiable at every point by claim 3 of Basic Properties of the Exponential Function and $\log$ at every point of $(0,\infty)$ by The Natural Logarithm; sums, products and constant multiples of two continuous maps are continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set, finite sums of constant multiples of continuous maps over a nonempty finite index set by Finite Linear Combinations of Continuous Real-Valued Maps on a Metric Space are Continuous §continuous, and compositions of continuous maps by claim 3 of Semicontinuity and Continuity Under Composition with a Continuous Map. So $x\mapsto K^{\downarrow}(x,A)$ is continuous, hence measurable with respect to $\mathcal{B}(X)$ by claim 3 of Borel Measurability and Bounded Integration on a Metric Space.

4. (Transport of laws)¶ For a probability measure $\lambda$ on $(\mathcal{C}_{M},2^{\mathcal{C}_{M}})$, $\lambda K^{\uparrow}_{J,\eta}$ is the map $B\mapsto(\lambda\otimes K^{\uparrow}_{J,\eta})(\mathcal{C}_{M}\times B)$ on $\mathcal{B}(X)$; for $\nu\in\mathcal{P}(X)$, $\nu K^{\downarrow}$ is the map $A\mapsto(\nu\otimes K^{\downarrow})(X\times A)$ on $2^{\mathcal{C}_{M}}$. Here $\lambda\otimes K^{\uparrow}_{J,\eta}$ and $\nu\otimes K^{\downarrow}$ are probability measures by Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral §composite, $\mathcal{C}_{M}\times B$ and $X\times A$ are measurable rectangles, and the maps $B\mapsto\mathcal{C}_{M}\times B$ and $A\mapsto X\times A$ send disjoint unions to disjoint unions and the whole factor to the whole product; so $\lambda K^{\uparrow}_{J,\eta}\in\mathcal{P}(X)$, and $\nu K^{\downarrow}$ is a probability measure on $(\mathcal{C}_{M},2^{\mathcal{C}_{M}})$. By the converse part of Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §measure, $\lambda=\lambda_{p}$ for $p(\sigma)=\lambda(\{\sigma\})$, and so, by Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral §composite and Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §integral,

$$(\lambda K^{\uparrow}_{J,\eta})(B)=\sum_{\sigma\in\mathcal{C}_{M}}\lambda(\{\sigma\})\,K^{\uparrow}_{J,\eta}(\sigma,B),\qquad(\nu K^{\downarrow})(A)=\int_{X}K^{\downarrow}(x,A)\,\nu(dx).$$

5. (The field law)¶ The Hubbard-Stratonovich field law of $\mathfrak{j}$ with shift $\eta$ is $\gamma^{\mathrm{HS}}_{J,\eta}=\mathbb{P}_{J}K^{\uparrow}_{J,\eta}\in\mathcal{P}(X)$, where $\mathbb{P}_{J}$ is a probability measure on $(\mathcal{C}_{M},2^{\mathcal{C}_{M}})$ by The Ising Measure on the Lattice Torus with a Reflection-Symmetric Periodic Pair Interaction §ising.

6. (The joint law)¶ The Hubbard-Stratonovich joint law is $\Xi_{J,\eta}=\mathbb{P}_{J}\otimes K^{\uparrow}_{J,\eta}$, a probability measure on $(\mathcal{C}_{M}\times X,\ 2^{\mathcal{C}_{M}}\otimes\mathcal{B}(X))$ by Integration Against a Probability Kernel: Measurable Sections, the Composite Measure on the Product and the Iterated Integral §composite.