Finite periodic Gaussian lattice field with positive mass

Assume the Axiom of Countable Choice from The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Let integers $d,L\ge 1$ specify the periodic lattice ${\mathrm{\Lambda }}_{d,L}=(\mathbb{Z}\mathrm{/}L\mathbb{Z}{)}^d$ and let $m^2>0$ and $\kappa >0$ be dimensionless real parameters in lattice units. On the real coordinate space ${\mathbb{R}}^{{\mathrm{\Lambda }}_{d,L}}$, let

$K_{d,L,m,\kappa }:=m^{2}I+\kappa (-\mathrm{\Delta }),$

where $-\mathrm{\Delta }$ is the periodic Laplacian of Fourier spectrum and positivity of the periodic lattice Laplacian. Its eigenvalues are

$m^2+4\kappa {\sum }_{j=1}^d{\sin }^2(\pi r_j\mathrm{/}L),r\in \{0,\dots ,L-1{\}}^d,$

so it is symmetric positive definite. The finite periodic Gaussian lattice field is the centered probability ${\gamma }_{K_{d,L,m,\kappa }}$ of Finite Gaussian probability with a positive precision matrix, with density proportional to $\exp (-z^{\mathsf{T}}{K}_{d,L,m,\kappa }z\mathrm{/}2)$ against Lebesgue measure on ${\mathbb{R}}^{L^d}$ and covariance $K_{d,L,m,\kappa }^{-1}$ by Gaussian covariance for positive quadratic forms and a boundary counterexample at $\beta =1$.

The lattice coordinates $z_x$ and the quadratic exponent are dimensionless. Under the physical quadratic convention of Finite Gaussian probability with a positive precision matrix, where $H(\mathrm{\Phi })=\frac{1}{2}{\mathrm{\Phi }}^{\mathsf{T}}{K}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{\Phi }$ and each field component has unit ${\mathrm{\Phi }}_{\ast }$, choosing $z_x={\mathrm{\Phi }}_x\mathrm{/}{\mathrm{\Phi }}_{\ast }$ gives dimensionless precision $K=\beta {\mathrm{\Phi }}_{\ast }^2{K}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ when the energy is divided by $k_{B}T$. Here $\beta =(k_{B}T{)}^{-1}$ with the SI units for $k_B$ and $T$ specified in Finite Gaussian probability with a positive precision matrix; equivalently the physical precision for this lattice model is $K_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=(k_{B}T\mathrm{/}{\mathrm{\Phi }}_{\ast }^2)K_{d,L,m,\kappa }$. This conversion makes no empirical claim. At $m^2=0$, the constant field is a zero mode and the unrestricted normalizer diverges by Unconstrained massless periodic Gaussian has infinite normalizer, so the full-space positive-precision definition does not apply. A finite-volume massless model must first specify a restriction such as the subspace ${\sum }_{x\in {\mathrm{\Lambda }}_{d,L}}{z}_x=0$ before inverting the Laplacian.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Fourier spectrum and positivity of the periodic lattice Laplacian

Unconstrained massless periodic Gaussian has infinite normalizer

Finite Gaussian probability with a positive precision matrix

Gaussian covariance for positive quadratic forms and a boundary counterexample