The Hubbard-Stratonovich Ising Field on the Lattice Torus: What the Lattice Results Model, the Role of the Choices, and the Temperature Regimes

This remark concerns the items around The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus, Well-Posedness of the Hamilton-Jacobi Equation of the Hubbard-Stratonovich Ising Field on the Lattice Torus at Every Cutoff, Interaction and Shift: Comparison, Existence and Uniqueness and The Gibbs Entropy Pair of the Hubbard-Stratonovich Ising Field is Lambda-Displacement Convex, with Lambda Uniform in the Cutoff when the Shifted Symbol Stays below One, with their notation: the lattice torus $\mathbb{L}_{M}$ of side $L=2M+1$, an interaction $\mathfrak{j}$ with symbol $\hat{J}$, a shift $\eta$ with $\mathfrak{q}_{k}=\hat{J}(k)+\eta>0$, the Ising measure $\mathbb{P}_{J}$ and the Hubbard-Stratonovich field law $\gamma^{\mathrm{HS}}_{J,\eta}$ on $X=H^{-m}(\mathbb{T}^{n})$.

1. (What is modelled.) A law of spin configurations is carried by finitely many points of any space into which configurations are embedded, so it is singular with respect to every diagonal Gaussian measure on $X$ with positive variances, and a penalty relative to a Gaussian reference cannot be applied to it directly. The Hubbard-Stratonovich transform replaces the spins by a field whose law has a density relative to a Gaussian (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 §density). The two kernels pass between the two descriptions: from a field to spins, the spins are sampled independently given the field; from spins to a field, the field is a smoothed magnetisation plus independent Gaussian noise. Both kernels contract relative entropy (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 §contraction-down and 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 §contraction-up). The Hamilton-Jacobi equation of The Hamilton-Jacobi Equation of the Hubbard-Stratonovich Ising Field on the Lattice Torus with White Noise is associated with the controlled Langevin dynamics of the field, whose invariant law is $\gamma^{\mathrm{HS}}_{J,\eta}$. It is not the Glauber spin-flip dynamics: that dynamics lives on the finite set of configurations and would need a transport geometry adapted to jumps, such as the one of Maas (arXiv:1102.5238v2).

2. (The choices.) The shift $\eta$ changes the field law but not the Ising measure, because $\sigma(z)^{2}=1$. The split $\varsigma$ of 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 is a device for meeting the lower bound required of an admissible potential, since $-\operatorname{lch}$ is not bounded below; the penalty pair and the equation do not depend on it (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 §independence). The embedding of lattice fields into distributions on the torus is not canonical. The one used here, $\mathcal{E}_{M}$, takes the discrete Fourier coefficients onto the Galerkin modes with the white-noise scaling $L^{n/2}$. In it, one unit of independent noise per site becomes white noise truncated to the modes of $\Gamma_{M}$, whose squared $L^{2}$ norm is the site sum (The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates §norm-identity), and the magnetisation field $L^{-n/2}\sum_{z}\sigma(z)\delta_{z}$, whose limit at high temperature is a nontrivial Gaussian field, has the same Fourier coefficients as $\mathcal{E}_{M}\sigma$ at every $k\in\Gamma_{M}$ and in addition nonzero coefficients outside $\Gamma_{M}$. Piecewise-constant interpolation, with the same scaling $L^{n/2}$, gives another embedding, whose coefficients agree with those of $\mathcal{E}_{M}$ approximately on modes $k$ with $\lVert k\rVert$ small compared with $L$.

3. (Temperature regimes.) Some admissible shift gives $\mathfrak{q}_{k}<1$ for every $k$ exactly when the largest value of $\hat{J}$ minus the smallest is less than $1$. This is the spectral condition of Bauerschmidt and Bodineau (arXiv:1712.03676v2), under which The Gibbs Entropy Pair of the Hubbard-Stratonovich Ising Field is Lambda-Displacement Convex, with Lambda Uniform in the Cutoff when the Shifted Symbol Stays below One §high-temperature gives displacement convexity with a constant uniform in the cutoff. For the nearest-neighbour interaction $\mathfrak{j}$ with inverse temperature $\beta$, the symbol $2\beta\sum_{i=1}^{n}\cos(2\pi k_{i}/L)$ takes values in $[-2n\beta,2n\beta]$, so $\beta<1/(4n)$ suffices; this is well below the critical temperature. For Kac interactions whose symbol is nonnegative, the width is at most $\hat{J}(0)$, the mean-field value, so the convex regime reaches the mean-field critical point; near it the rescaled field converges to the $\Phi^{4}$ field in dimension $2$ (Mourrat and Weber, arXiv:1410.1179v2) and in dimension $3$ (Grazieschi, Matetski and Weber, arXiv:2303.10242v2), and the shift of the critical temperature is of the order of the Wick constant. Between the convex regime and the critical point, the log-Sobolev inequality of Bauerschmidt and Dagallier (arXiv:2202.02301v2) remains available, while The Gibbs Entropy Pair of the Hubbard-Stratonovich Ising Field is Lambda-Displacement Convex, with Lambda Uniform in the Cutoff when the Shifted Symbol Stays below One §high-temperature no longer applies.

4. (What is not proved.) Nothing here concerns the limit $M\to\infty$. The Gaussian references $\gamma_{c^{J,\eta}}$ depend on $M$, and under the rescaling of a scaling limit they converge only weakly and become mutually singular with the limiting reference, so the noise geometry moves with the cutoff. Removing the cutoff therefore needs a stability result for moving reference measures.