[Paper Review] The Discrete Gaussian model, II. Infinite-volume scaling limit at high temperature

Roland Bauerschmidt, Jiwoon Park+1 more|arXiv (Cornell University)|Feb 4, 2022

Stochastic processes and statistical mechanics40 references4 citations

TL;DR

This paper establishes the infinite-volume scaling limit of the two-dimensional Discrete Gaussian model at high temperature, showing it converges to a multiple of the Gaussian free field. Using a renormalisation group approach with general external fields, the authors prove the scaling limit for both infinite-volume gradient Gibbs states and mesoscopic test functions on the torus, extending prior results on macroscopic test functions.

ABSTRACT

The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integer-valued. In two dimensions, at sufficiently high temperature, we show that the scaling limit of the infinite-volume gradient Gibbs state with zero mean is a multiple of the Gaussian free field. This article is the second in a series on the Discrete Gaussian model, extending the methods of the first paper by the analysis of general external fields (rather than macroscopic test functions on the torus). As a byproduct, we also obtain a scaling limit for mesoscopic test functions on the torus.

Motivation & Objective

• To establish the infinite-volume scaling limit of the Discrete Gaussian model in two dimensions at high temperature.
• To extend the scaling limit results from macroscopic to general external fields, including mesoscopic test functions on the torus.
• To prove the convergence of the infinite-volume gradient Gibbs state to a multiple of the Gaussian free field.
• To develop a renormalisation group framework capable of handling general external fields beyond macroscopic test functions.
• To provide a rigorous derivation of the scaling limit using a novel reblocking and fluctuation integral decomposition.

Proposed method

• Uses a renormalisation group map with external fields, generalizing the approach from the first paper to handle non-macroscopic test functions.
• Applies a reblocking procedure to decompose the external field into scale-dependent components, enabling iterative analysis across multiple scales.
• Employs a fluctuation integral decomposition to separate high- and low-energy contributions in the field configuration.
• Introduces a regulator with external field to control the growth of correlation functions under scale iteration.
• Uses a block decomposition of the field space and iteratively controls the effective action through contraction estimates in a normed space.
• Applies a multiscale expansion to express the partition function as a sum over block configurations, with explicit control over the error terms.

Experimental results

Research questions

• RQ1 Does the infinite-volume gradient Gibbs state of the 2D Discrete Gaussian model converge to a multiple of the Gaussian free field at high temperature?
• RQ2 Can the scaling limit be extended from macroscopic test functions to general external fields, including mesoscopic scales on the torus?
• RQ3 What is the role of the external field in the renormalisation group framework for the Discrete Gaussian model?
• RQ4 How does the presence of general external fields affect the convergence of the scaling limit?
• RQ5 Is the scaling limit robust under finite-range interactions and zero-mean gradient conditions?

Key findings

• The infinite-volume gradient Gibbs state of the 2D Discrete Gaussian model at high temperature converges weakly to a multiple of the Gaussian free field.
• The scaling limit holds for general external fields, not restricted to macroscopic test functions, enabling the analysis of mesoscopic observables on the torus.
• The convergence is established via a multiscale renormalisation group approach with explicit control over the effective action and error terms.
• The authors prove the existence of the infinite-volume limit through weak convergence with periodic boundary conditions on tori of side length $2^N$.
• The scaling limit is shown to be universal in the sense that it depends only on the asymptotic logarithmic decay of the Green's function, $(- abla_J)^{-1}(x,y) \sim -\frac{1}{2\pi v_J^2}\log|x-y|$.
• The result confirms that the discrete nature of the spins does not obstruct the Gaussian free field scaling limit at high temperature, even in the infinite-volume limit.