The Discrete Gaussian model, I. Renormalisation group flow at high temperature

Abstract

The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integervalued. In two dimensions, at sufficiently high temperature, we show that its macroscopic scaling limit on the torus is a multiple of the Gaussian free field. Our proof starts from a single renormalisation group step after which the integer-valued field becomes a smooth field which we then analyse using the renormalisation group method.This paper also provides the foundation for the construction of the scaling limit of the infinite-volume gradient Gibbs state of the Discrete Gaussian model in the companion paper. Moreover, we develop all estimates for general finite-range interaction with sharp dependence on the range. We expect these estimates to prepare for a future analysis of the spread-out version of the Discrete Gaussian model at its critical temperature.

Weakly non-planar dimers

“…Recently, remarkable progress has been made on (logarithmic) delocalization of discrete, two-dimensional random interfaces. We start with the result which is maybe the closest in spirit to our work, that is [2,3]. These works prove, by means of bosonic, constructive RG methods, that the height function of the discrete Gaussian interface model (that is the lattice GFF conditioned to be integer-valued) has, at sufficiently high temperature, admits the continuum GFF as scaling limit.…”

“…(3.47) 7 We refer e.g. to [25,Remark 5] for the meaning of the derivative with respect to Grassmann variables Three cases will play a central role in the following: the interacting propagator G (2) , the interacting vertex function G (2,1) and the interacting dimer-dimer correlation G (0,2) , which deserve a distinguished notation: letting x = (x, ), y = (y, ), z = (z, ), and denoting by e (resp. e ) the edge with black vertex x = (x, ) (resp.…”

“…. , y n ) exist; we denote the limit simply by dropping the index L. Let us define the Fourier transforms of the interacting propagator and interacting vertex function via the following conventions: for , , ∈ I and j ∈ J , we let Ĝ (2) , (p) :=…”

“…. , η y (k) ∈ F , with y (1) = x (1) , y (k) = x (2) and |y (a) − y (a+1) | = 1. The set of edges crossed by the path between η y (a) and η y (a+1) , denoted E (a) , is a translation of either E 1,0 (if y (a+1) − y (a) is horizontal) or E 2,0 (if y (a+1) − y (a) is vertical).…”

A central limit theorem for square ice

“…The seminal work of Fröhlich and Spencer [11] established the logarithmic fluctuation of the Discrete Gaussian and the absolute value potential model. There are extensive recent results on solid-on-solid models: the GFF scaling limit for the Discrete Gaussian case was proved in [2,3], and the (qualitative) delocalization were established for general potential [16,17,1,22].…”