[Paper Review] Maximum of the integer-valued Gaussian free field

Mateo Wirth|arXiv (Cornell University)|Jul 20, 2019

Stochastic processes and statistical mechanics12 references4 citations

TL;DR

This paper establishes that the maximum of the integer-valued Gaussian free field (IV-GFF) on a two-dimensional box of side length $L$ grows logarithmically with $L$ at high temperature, specifically of order $\log L$. Using techniques from the Fröhlich-Spencer proof and spin wave constructions, it proves that the maximum is stochastically bounded below by $c_0/\sqrt{\beta} \log L$ with high probability, matching the order of the standard discrete GFF and confirming delocalization at high temperature.

ABSTRACT

We investigate the order of the maximum of the integer-valued Gaussian free field in two dimensions, and show that it grows logarithmically with the size of the box. Our treatment follows closely that of a recent paper by Kharash and Peled on the Fröhlich-Spencer proof of the Berezinskii-Kosterlitz-Thouless transition.

Motivation & Objective

• To understand the fine-scale statistical behavior of the integer-valued Gaussian free field (IV-GFF), particularly its maximum value in two dimensions.
• To establish that the maximum of the IV-GFF is of order $\log L$ at high temperature, matching the behavior of the continuous GFF.
• To extend the Fröhlich-Spencer delocalization argument to the integer-valued case by proving sharp lower bounds on the maximum.
• To demonstrate that the IV-GFF exhibits delocalization through the logarithmic growth of its maximum, confirming predictions from topological phase theory.

Proposed method

• Adapts the Fröhlich-Spencer proof framework to the integer-valued setting using a symmetrized version of the IV-GFF.
• Employs a moment generating function lower bound for the symmetrized IV-GFF, generalizing a result from [6] to arbitrary boundary conditions.
• Applies the Markov field property of the IV-GFF to decompose the domain $\Lambda$ into subdomains for independent analysis.
• Uses spin wave constructions—functions $a_{s,\rho^*}$ with localized support and bounded energy—for each square in a hierarchical decomposition of the domain.
• Combines energy contributions from multiple spin waves to derive a lower bound on the total energy, which controls the typical size of the field.
• Applies the Borel-Cantelli lemma and union bounds across subdomains to show that the maximum exceeds $c_1/\sqrt{\beta} \log L$ with high probability.

Experimental results

Research questions

• RQ1 What is the asymptotic order of the maximum of the integer-valued Gaussian free field on a two-dimensional box as the system size $L \to \infty$?
• RQ2 Does the IV-GFF exhibit delocalization at high temperature, and if so, what is the scaling of its maximum?
• RQ3 Can the techniques used to analyze the continuous GFF be adapted to prove analogous results for the integer-valued version?
• RQ4 How do spin wave constructions contribute to lower bounding the maximum of the IV-GFF?
• RQ5 Is the maximum of the IV-GFF stochastically bounded below by a multiple of $\log L$, and what is the dependence on the inverse temperature $\beta$?

Key findings

• The maximum of the IV-GFF satisfies $\mathbb{P}^\text{IV}_{\beta,\Lambda,\mathbf{0}}\left(\max_{j\in\Lambda}|m_j| \geq \frac{c_0}{\sqrt{\beta}} \log L\right) \geq 1 - L^{-\eta_0}$ for large $L$, proving a lower bound of order $\log L$.
• With high probability, $\max_{j\in\Lambda} m_j \geq \frac{c_1}{\sqrt{\beta}} \log L$, confirming that the maximum is of order $\log L$ for $\beta < \beta_0$.
• The lower bound is sharp, as the maximum is known to be at most $O(\log L)$, so the order is exactly $\log L$.
• The proof relies on a lower bound on the moment generating function of the symmetrized IV-GFF, which generalizes a result from [6] to arbitrary boundary conditions.
• Spin wave functions $a_{s,\rho^*}$ with localized support and energy $\geq D_5/\beta$ are constructed to control the field's fluctuations and ensure non-trivial lower bounds.
• The energy contribution from multiple spin waves is additive and non-overlapping, allowing the construction of a field with large maximum value with high probability.