Macroscopic freezing of a vector Gaussian free field conditioned to avoid a ball

Technical Summary: Macroscopic Freezing of a Vector Gaussian Free Field Conditioned to Avoid a Ball

Problem Statement
The paper investigates the macroscopic behavior of a vector-valued discrete Gaussian free field (GFF) on a two-dimensional lattice box $\Lambda_n$ of side length $n$, conditioned to stay outside a fixed ball of radius $R$ on a macroscopic interior domain $D_n$. Specifically, the author considers $m \ge 2$ independent zero-boundary GFF coordinates $\phi_n = (\phi_n^1, \dots, \phi_n^m)$. The conditioning event $\Omega_n$ requires that the Euclidean norm of the field vector $|\phi_n(x)| \ge R$ for all $x \in D_n$.

The central question addresses "entropic repulsion" in the vector setting: while scalar fields are forced to lift away from an obstacle, a vector field has a free direction. The paper seeks to determine whether different parts of the macroscopic domain $D$ choose the same direction (alignment) or if the direction varies spatially. This builds upon previous work by Korzhenkova and Sepúlveda [1], who established that the norm of the field is typically $2 \log n$ and that directions align at sub-macroscopic distances ($n^\nu$ for $\nu < 1$), but left open the question of "macroscopic freezing" (alignment across the entire domain).

Methodology
The proof combines three primary technical components:

1. Norm Repulsion Theorems: The author utilizes results from Korzhenkova and Sepúlveda [1] to establish that, under the conditioning, the normalized field $|\phi_n(x)|/\log n$ concentrates around 2 almost everywhere on $D$. This is formalized via a bounded continuous functional $G(f)$ measuring the $L^2$ deficit from the target norm 2.
2. Gaussian Large Deviations: The paper employs an $L^2$ large deviation principle (LDP) for the unconditioned field $X_n = \phi_n / \log n$. The rate function is identified as the Dirichlet energy $I(f) = \frac{1}{4\pi} \int_\Lambda |\nabla f|^2$. The conditioning event is analyzed by restricting the LDP to the set where the norm constraint is satisfied.
3. Vector Capacity Variational Problem: The core of the argument reduces the vector problem to a scalar one. The author analyzes the minimization of the rate function $I(f)$ subject to the constraint $|f| \ge 2$ on $D$. They prove that the minimizers of this vector capacity problem are unique up to a global rotation: the field must take the form $f = 2 U h_D$, where $h_D$ is the equilibrium potential of the domain $D$ (relative to $\Lambda$) and $U$ is a constant unit vector.

A crucial step involves showing that any variation in the direction vector $s(u) = f(u)/|f(u)|$ incurs an additional "angular energy" cost proportional to $\int h_D^2 |\nabla s|^2$. Since the domain $\Lambda$ is connected, the energy minimization forces $s(u)$ to be constant almost everywhere, thereby establishing macroscopic alignment.

Key Results
The paper establishes the following main results:

• Asymptotic Cost of Conditioning (Theorem 1.1): The probability of the conditioning event decays as:
  $$\lim_{n \to \infty} \frac{\log P(\Omega_n)}{(\log n)^2} = -\frac{2}{\pi} \text{cap}_\Lambda(D)$$
  where $\text{cap}_\Lambda(D)$ is the relative capacity of $D$ in $\Lambda$. This cost is independent of the ball radius $R$ and the vector dimension $m$.

• Macroscopic Convergence (Theorem 1.1): Under the conditional measure $Q_n$, the normalized field $X_n$ converges in distribution in $L^2(\Lambda; \mathbb{R}^m)$ to $2 U h_D$, where $U$ is a random unit vector uniformly distributed on the sphere $S^{m-1}$. This implies that the field profile is deterministic up to a global random rotation.

• Deterministic Alignment (Corollary 1.2): There exists a deterministic subset of sites $\tilde{D}_n \subset D_n$ with density approaching 1 (i.e., $|D_n \setminus \tilde{D}_n| = o(n^2)$) such that the expected alignment between any two sites in $\tilde{D}_n$ converges to 1:
  $$\inf_{x,y \in \tilde{D}_n} E[\sigma_n(x) \cdot \sigma_n(y) \mid \Omega_n] \to 1$$
  where $\sigma_n(x)$ is the direction of the field at $x$. This confirms that the "freezing" of the direction occurs across the entire macroscopic domain, not just locally.

Significance and Scope
The paper provides a complete spatial profile for the entropic repulsion of a vector GFF, resolving the question of macroscopic freezing posed in [1]. The significance lies in demonstrating that the constraint of avoiding a ball forces the vector field to select a single, global direction to minimize the Dirichlet energy, effectively coupling the directions chosen in potentially disconnected components of the domain $D$.

The author notes that their approach relies on the equality case in the capacity variational problem and the spectral properties of the GFF. They explicitly state that the results do not address:

• Quantitative rates of profile concentration.
• The removal of the exceptional set in Corollary 1.2 for fixed interior subdomains.
• The finer-scale fluctuations of the residual field $X_n - 2 U h_D$.
• The next-order asymptotic terms that might depend on the radius $R$ or dimension $m$.

The work is framed as a rigorous extension of classical scalar entropic repulsion results (e.g., Ben Arous and Deuschel [2]) to the vector setting, utilizing the interplay between large deviations and capacity theory.