[Paper Review] Quantitative bounds on vortex fluctuations in $2d$ Coulomb gas and maximum of the integer-valued Gaussian free field

Christophe Garban, Avelio Sepúlveda|arXiv (Cornell University)|Dec 2, 2020

Theoretical and Computational Physics41 references4 citations

TL;DR

This paper establishes the first quantitative non-perturbative lower bounds on vortex-induced fluctuations in 2D Coulomb gases and related models like the Villain model and integer-valued Gaussian free field (GFF). By introducing a novel non-perturbative sampling method via a conditional edge-mass representation, it proves that vortex fluctuations are at least as strong as spin-wave fluctuations at high inverse temperature, and derives explicit upper bounds on the two-point correlation function and the maximum of the integer-valued GFF.

ABSTRACT

In this paper, we study the influence of the vortices on the fluctuations of $2d$ systems such as the Coulomb gas, the Villain model or the integer-valued Gaussian free field. In the case of the $2d$ Villain model, we prove that the fluctuations induced by the vortices are at least of the same order of magnitude as the ones produced by the spin-wave. We obtain the following quantitative upper-bound on the two-point correlation in $\\mathbb{Z}^2$ when $\\beta>1$ \\[ \\langle\\sigma_x \\sigma_y\ angle_{\\beta}^{Villain} \\leq C \\, \\left( \\frac 1 {\\|x-y\\|_2}\ ight)^{\\frac 1 {2\\pi \\beta}\\left ( 1+\\beta e^{-\\frac{(2\\pi)^2}{2} \\beta}\ ight )} \\] The proof is entirely non-perturbative. Furthermore it provides a new and algorithmically efficient way of sampling the $2d$ Coulomb gas. For the $2d$ Coulomb gas, we obtain the following lower bound on its fluctuations at high inverse temperature \\[ \\mathbb{E}_\\beta^{Coul}[\\langle \\Delta^{-1}q, g\ angle] \\geq \\exp(-\\pi^2 \\beta + o(\\beta)) \\langle g,(-\\Delta)^{-1}g \ angle \\] This estimate coincides with the predictions based on a RG analysis from [JKKN77] and suggests that the Coulomb potential $\\Delta^{-1}q$ at inverse temperature $\\beta$ should scale like a Gaussian free field of inverse temperature of order $\\exp(\\pi^2 \\beta)$. Finally, we transfer the above vortex fluctuations via a duality identity to the integer-valued GFF by showing that its maximum deviates in a quantitative way from the maximum of a usual GFF. More precisely, we show that with high probability when $\\beta>1$ \\[ \\max_{x\\in [-n,n]^2} \\Psi_n(x) \\leq \\sqrt{\\frac{2\\beta}{\\pi} \\big(1 - \\beta e^{- \\frac{(2\\pi)^2\\beta} {2} } \\big)} \\log n \\,. \\] where $\\Psi_n$ is an integer-valued GFF in the box $[-n,n]^2$ at inverse temperature $\\beta^{-1}$. Applications to the free-energies of the Coulomb gas, the Villain model and the integer-valued GFF are also considered.

Motivation & Objective

• To quantify the contribution of vortices (topological defects) to macroscopic fluctuations in 2D statistical mechanics models such as the Coulomb gas, Villain model, and integer-valued GFF.
• To close the gap in existing literature by providing the first non-perturbative lower bounds on vortex-induced fluctuations, which were previously inaccessible via standard perturbative or moment-based methods.
• To develop a new algorithmic framework for efficiently sampling the 2D Coulomb gas using local Markov chain Monte Carlo updates, overcoming the challenge of long-range interactions.
• To establish a duality between vortex fluctuations in the Villain model and the maximum of the integer-valued GFF, quantifying how topological defects affect extreme value statistics.
• To validate and refine predictions from the renormalization group analysis of [JKKN77] by deriving matching quantitative bounds in the low-temperature regime.

Proposed method

• Introduces a conditional sampling scheme where, for a given spin configuration in the Villain model, edge variables $ m_e \in \mathbb{Z} $ are sampled independently on each edge such that the resulting charge $ q = \mathbf{d}m $ captures the fluctuations of the 2D Coulomb gas.
• Uses the discrete exterior derivative $ \mathbf{d} $ to define the vortex charge $ q $, enabling a direct link between spin configurations and the Coulomb gas measure.
• Applies Ginibre's correlation inequality and convergence results from [MMSP+78] to compare finite-volume measures with the infinite-volume Gibbs measure on $ \mathbb{Z}^2 $, ensuring consistency under boundary conditions.
• Derives a quantitative bound on the probability that the gradient $ \mathbf{d}\theta(e) $ deviates from zero, using a Gaussian approximation and exponential moment bounds.
• Transfers results from the Villain model to the integer-valued GFF via a duality identity, relating the maximum of the integer-valued field to the Coulomb gas's charge fluctuations.
• Employs a non-perturbative approach based on large-deviation estimates and exponential moment bounds, avoiding reliance on perturbation theory or RG approximations.

Experimental results

Research questions

• RQ1 How do vortices contribute quantitatively to the macroscopic fluctuations in 2D Coulomb gases and related models at high inverse temperature?
• RQ2 Can non-perturbative methods provide lower bounds on vortex fluctuations where perturbative techniques fail?
• RQ3 What is the precise scaling of the two-point correlation function in the 2D Villain model when vortex contributions are included?
• RQ4 How does the maximum of the integer-valued Gaussian free field deviate from that of the standard GFF due to vortex-induced fluctuations?
• RQ5 Can a local MCMC algorithm be constructed for sampling the 2D Coulomb gas by exploiting the conditional independence of vortex charges?

Key findings

• For the 2D Villain model at inverse temperature $ \beta > 1 $, the two-point correlation function satisfies the upper bound $ \langle \sigma_x \sigma_y \rangle_{\beta}^{\text{Villain}} \leq C \left( \frac{1}{\|x-y\|_2} \right)^{\frac{1}{2\pi\beta}\left(1 + \beta e^{-\frac{(2\pi)^2}{2}\beta}\right)} $, showing vortex fluctuations are of the same order as spin-wave fluctuations.
• For the 2D Coulomb gas at high $ \beta $, the expected value of the potential $ \langle \Delta^{-1}q, g \rangle $ satisfies $ \mathbb{E}_{\beta}^{\text{Coul}}[\langle \Delta^{-1}q, g \rangle] \geq \exp(-\pi^2\beta + o(\beta)) \langle g, (-\Delta)^{-1}g \rangle $, matching predictions from [JKKN77] and suggesting effective inverse temperature $ \sim \exp(\pi^2\beta) $.
• The maximum of the integer-valued GFF $ \Psi_n $ on $ [-n,n]^2 $ satisfies $ \max_{x \in [-n,n]^2} \Psi_n(x) \leq \sqrt{\frac{2\beta}{\pi}\left(1 - \beta e^{-\frac{(2\pi)^2\beta}{2}}\right)} \log n $ with high probability when $ \beta > 1 $, quantifying the suppression due to vortices.
• The proposed method enables a new, algorithmically efficient MCMC sampling scheme for the 2D Coulomb gas using only local updates, bypassing the need for non-local updates due to long-range interactions.
• The paper establishes that vortex fluctuations are not negligible in the low-temperature regime and are responsible for a significant portion of the total fluctuation, resolving a long-standing open question about their contribution.
• The duality between the Villain model and the integer-valued GFF is made quantitative, showing that vortex-induced fluctuations reduce the maximum of the field by a factor depending on $ \beta $, consistent with renormalization group expectations.