Invisibility of the integers for the discrete Gaussian chain via a Caffarelli-Silvestre extension of the discrete fractional Laplacian

Abstract

The Discrete Gaussian Chain is a model of interfaces $\Psi :\mathbf{Z}\to \mathbf{Z}$ governed by the Hamiltonian $$H(\Psi )=\sum _{i\ne j}{J}_{\alpha }(|i-j|)|{\Psi }_i-{\Psi }_j{|}^2$$ with long-range coupling constants $J_{\alpha }(k)\asymp k^{-\alpha }$. For any $\alpha \in [2,3)$ and at high enough temperature, we prove an invariance principle for such an $\alpha$-Discrete Gaussian Chain towards a $H(\alpha )$-fractional Gaussian process where the Hurst index $H$ satisfies $H=H(\alpha )=\frac{\alpha -2}{2}$. This result goes beyond a conjecture by Fröhlich and Zegarlinski [FZ91] which conjectured fluctuations of order $n^{\frac{1}{2}(\alpha -2)\wedge 1}$ for the Discrete Gaussian Chain. More surprisingly, as opposed to the case of the $2D$ Discrete Gaussian $\Psi :{\mathbf{Z}}^2\to \mathbf{Z}$, we prove that the integers do not affect the {\em effective temperature} of the discrete Gaussian Chain at large scales. Such an {\em invisibility of the integers} had been predicted by Slurink and Hilhorst in the special case ${\alpha }_c=2$ in [SH83]. We also identify a similar invisibility of integers when a $2D$ Gaussian Free Field at high temperature is conditioned to take integer values on a dilute enough "fractal subset" of ${\mathbf{Z}}^2$. Our proof relies on four main ingredients: (1) A Caffareli-Silvestre extension for the discrete fractional Laplacian (which may be of independent interest) (2) A localisation of the chain in a smoother sub-domain (3) A Coulomb gas-type expansion in the spirit of Fröhlich-Spencer [FS82] (4) Controlling the amount of Dirichlet Energy supported by a $1D$ band for the Green functions of ${\mathbf{Z}}^2$ Bessel-type random walks Finally, we also analyse the (easier) regime $\alpha \in (1,2)\cup (3,\infty )$ as well as the $2D$ Discrete Gaussian with long-range coupling constants (for any $\alpha >{\alpha }_c=4$).

65 pages