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arxiv: 2509.21538 · v1 · pith:TTPQBPWXnew · submitted 2025-09-25 · 🧮 math.PR · math-ph· math.MP

Vector-valued Gaussian free field conditioned to avoid a ball: Entropic repulsion of the norm and Freezing of spins

Aleksandra Korzhenkova , Avelio Sep\'ulveda This is my paper

Pith reviewed 2026-05-21 21:56 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords Gaussian free fieldvector-valuedconditioningentropic repulsionfreezingmassive Gaussian fieldphase transition
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The pith

When conditioned to avoid a ball, the vector-valued Gaussian free field's norm repels entropically and its angles freeze at mesoscopic scales.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies two-dimensional vector-valued Gaussian free fields and their massive lattice versions conditioned to avoid a ball at every site in a subdomain. For the massless Dirichlet case it shows that the norm pushes away from zero through entropic repulsion while the vector directions lock in place over all medium length scales. The argument first establishes that the unconditioned field fills its range without holes near typical interior points. In the massive setting the conditioned norm stays bounded as the domain grows, which permits infinite-volume Gibbs measures. The scalar massive version further exhibits a phase transition: small avoided intervals produce a unique infinite-volume limit while large centered intervals allow multiple such limits.

Core claim

Under conditioning to avoid a ball at every site, the norm of the massless vector-valued Gaussian free field exhibits entropic repulsion while angular components freeze at mesoscopic scales. This relies on showing the unconditioned field has no holes around bulk points. In the massive case the norm is uniformly bounded, enabling infinite-volume Gibbs measures. The scalar massive case shows a phase transition depending on avoided interval size, with unique or multiple limits.

What carries the argument

The no-holes property of the unconditioned vector-valued field around bulk range points, which allows the conditioning to induce norm repulsion and angular freezing without voids disrupting the behavior.

Load-bearing premise

The unconditioned vector-valued field has no holes around points in the bulk of the range.

What would settle it

Finding that the unconditioned field has holes around a bulk point with positive probability would disprove the key step needed for the repulsion and freezing conclusions.

Figures

Figures reproduced from arXiv: 2509.21538 by Aleksandra Korzhenkova, Avelio Sep\'ulveda.

Figure 1
Figure 1. Figure 1: The part of the blue subgrid of Z 2 fully contained in Dn is ⌗α(x0). The union of boxes B formed by this part of the subgrid is Πα(x0). the equations involving, say, both Pn and Pn,N the former stands for the law of the scalar field, while the latter denotes the law of the N-vectorial field. 2.2 Correlations and FKG inequality The goal of this section is to recall and prove some technical results related t… view at source ↗
Figure 2
Figure 2. Figure 2: The part of the blue subgrid of Z 2 contained in Dn is ⌗α(x0). The union of boxes B formed by this subgrid is Πα(x0). The thick blue lines form Iα, the interface between positive and negative boxes contained in Dn. The blue boxes are exactly the ones counted in Nα −|Iα = Nα <−δ |Iα + Nα −δ,0 |Iα . Furthermore, for N ∈ {Nα −, Nα <−δ , Nα −δ,0 }, we denote by N|Iα the number of boxes of the respective type, … view at source ↗
read the original abstract

We study the laws of the two-dimensional vector-valued Dirichlet Gaussian free field and its massive lattice counterpart, conditioned to avoid a ball at every site of a subdomain. We prove that, under this conditioning, the norm of the massless field exhibits entropic repulsion, while its angular components freeze at all mesoscopic scales. A key step in the analysis is showing that around any given point in the bulk of the range, the unconditioned field has no holes. In the massive case, the conditioned field behaves differently: its norm remains uniformly bounded as the system size grows, leading to the existence of infinite-volume Gibbs measures. Furthermore, in the scalar massive case, the system undergoes a phase transition in the size of the avoided interval: for small intervals, the system admits a unique infinite-volume limit, while for sufficiently large centered intervals, multiple such limits exist.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the two-dimensional vector-valued Dirichlet Gaussian free field (and its massive lattice counterpart) conditioned to avoid a ball at every site of a subdomain. It claims to prove that the norm of the massless field exhibits entropic repulsion while its angular components freeze at all mesoscopic scales, with the key step being a demonstration that the unconditioned field has no holes around any point in the bulk of the range. In the massive case the conditioned norm remains uniformly bounded (yielding infinite-volume Gibbs measures), and the scalar massive case is shown to undergo a phase transition in the size of the avoided interval, with uniqueness of the infinite-volume limit for small intervals and non-uniqueness for sufficiently large centered intervals.

Significance. If the central claims hold, the work would advance the understanding of conditioned vector-valued Gaussian free fields by establishing entropic repulsion and angular freezing phenomena, together with a clear separation between massless and massive regimes and a phase transition in the scalar massive setting. The explicit identification of the no-holes property of the unconditioned field as an independent enabling step is a structural strength that could facilitate extensions to other conditioning events.

major comments (1)
  1. [Section containing the proof of the no-holes property (referenced in the abstract as the key step)] The no-holes property of the unconditioned vector-valued field (around bulk points) is explicitly identified as the key step enabling both the entropic-repulsion and angular-freezing conclusions under ball-avoidance conditioning. The manuscript must supply the full derivation, including all error controls and the precise manner in which the property transfers to the conditioned law; without these details the load-bearing step cannot be verified.
minor comments (2)
  1. [Introduction / Notation subsection] Notation for the vector-valued field and the avoided ball should be introduced with a single consistent definition early in the text rather than piecemeal.
  2. [Theorem on scalar massive phase transition] The statement of the phase-transition threshold for the scalar massive case would benefit from an explicit numerical or scaling criterion that distinguishes the small-interval and large-interval regimes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive overall assessment and for the constructive major comment. We respond to it below and will incorporate the requested clarifications in a revised version of the manuscript.

read point-by-point responses

  1. Referee: [Section containing the proof of the no-holes property (referenced in the abstract as the key step)] The no-holes property of the unconditioned vector-valued field (around bulk points) is explicitly identified as the key step enabling both the entropic-repulsion and angular-freezing conclusions under ball-avoidance conditioning. The manuscript must supply the full derivation, including all error controls and the precise manner in which the property transfers to the conditioned law; without these details the load-bearing step cannot be verified.

    Authors: We agree that a fully self-contained derivation of the no-holes property, with explicit error controls and a precise account of its transfer to the conditioned measure, is necessary for verification. The current manuscript contains the argument in the section referenced in the abstract, relying on Green's function estimates and Gaussian tail bounds for the vector field. In the revision we will expand this section to include all intermediate error estimates with explicit constants and scale ranges, together with a dedicated paragraph detailing the change-of-measure argument that transfers the no-holes property to the ball-avoidance conditioning. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes its main results on entropic repulsion and angular freezing for the conditioned vector-valued GFF by first proving an independent property of the unconditioned field: that it has no holes around bulk points. This property is not defined in terms of the conditioned object or derived from it by construction. The massive-case analysis and phase-transition claims follow from separate estimates on infinite-volume limits. No load-bearing step reduces to a self-definition, a fitted input renamed as prediction, or a self-citation chain that replaces external verification. The argument structure is consistent with standard rigorous probabilistic derivations that rely on unconditioned properties as inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Full text unavailable; abstract does not list explicit free parameters, axioms, or invented entities. The work appears to rely on standard properties of Gaussian free fields and Dirichlet boundary conditions.

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