Persistence and entropic repulsion of stationary Gaussian fields with spectral singularity at the origin
Pith reviewed 2026-05-21 03:06 UTC · model grok-4.3
The pith
Stationary Gaussian fields with spectral singularity of order alpha at the origin have persistence probabilities whose log-asymptotics are universal in alpha and dimension d, given explicitly by the capacity of the alpha-Riesz kernel.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For a d-dimensional stationary Gaussian field whose spectral measure has a singularity of order alpha at the origin, the logarithm of the persistence probability over an interval of length T admits an exact asymptotic formula determined solely by alpha and d; the formula is expressed through the capacity and equilibrium potential of the alpha-Riesz kernel. The same objects describe the entropic repulsion profile of the field conditioned on persistence. Both statements hold under mild regularity assumptions on the covariance or spectral measure and generalize the corresponding result for the Gaussian free field.
What carries the argument
The capacity and equilibrium potential of the alpha-Riesz kernel, which convert the local spectral singularity into global quantitative control on the probability of staying positive and on the conditioned field shape.
If this is right
- The persistence probability decays at an explicit rate fixed by the Riesz capacity of the singularity order alpha.
- The typical realization conditioned on persistence is repelled from zero according to the equilibrium potential of the same kernel.
- Both quantities are insensitive to further details of the covariance once the regularity conditions hold.
- The same explicit formulas apply to any field in the broad class, including but not limited to the Gaussian free field.
Where Pith is reading between the lines
- The universality may extend to certain non-Gaussian fields whose local fluctuations near zero mimic the Gaussian case.
- Numerical sampling of fields with controlled spectra could directly verify the predicted dependence on alpha and d.
- The Riesz-capacity description suggests analogies with electrostatics or potential theory that might simplify calculations in related models of interfaces.
Load-bearing premise
The covariance or spectral measure satisfies mild regularity conditions that make the asymptotics depend only on alpha and d rather than on finer details of the field.
What would settle it
Construct a stationary Gaussian field whose spectral measure has the same singularity of order alpha at zero but violates the regularity conditions, then compute or simulate its persistence probability and check whether the log-asymptotics match the Riesz-capacity formula.
read the original abstract
We compute the exact log-asymptotics of the persistence probability, and determine the entropic repulsion profile conditioned on persistence, for general $d$-dimensional stationary Gaussian fields with spectral singularity at the origin of order $\alpha \in [0,d)$. Under mild regularity conditions these are shown to be universal, depending only on $\alpha$ and $d$, and to have explicit formulations in terms of the capacity and equilibrium potential of the $\alpha$-Riesz kernel. This generalises a result of Bolthausen, Deuschel and Zeitouni on the Gaussian free field to a wide class of Gaussian fields with spectral singularity.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to compute the exact log-asymptotics of the persistence probability and the entropic repulsion profile for general d-dimensional stationary Gaussian fields with a spectral singularity of order α at the origin (α ∈ [0,d)). Under mild regularity conditions on the covariance or spectral measure, these quantities are universal, depending only on α and d, and are given explicitly in terms of the capacity and equilibrium potential of the α-Riesz kernel. This generalizes the result of Bolthausen, Deuschel and Zeitouni for the Gaussian free field.
Significance. The results are significant as they provide a universal description for persistence and conditioned large deviations in a broad class of Gaussian fields, reducing the problem to classical potential theory via the α-Riesz kernel. This extends previous work on the GFF and offers explicit formulas that depend only on the singularity order, which is valuable for applications in probability and physics. The approach using Fourier analysis to separate the singular spectral contribution is technically sound and generalizes without relying on Markov properties.
minor comments (2)
- [Abstract] The abstract refers to 'mild regularity conditions' without a brief indication of their form (e.g., density ~ |ξ|^{-α} near zero with integrability away from zero); adding one sentence would improve accessibility.
- [Theorem 1.1] In the statement of the main theorem, the precise range of α relative to d could be cross-referenced to the spectral assumption in Section 2 to avoid any ambiguity for readers.
Simulated Author's Rebuttal
We thank the referee for their positive summary and assessment of the significance of our results. The referee correctly identifies that our work provides a universal description via the α-Riesz kernel, generalizing the Bolthausen-Deuschel-Zeitouni result on the Gaussian free field. We appreciate the recommendation for minor revision. No specific major comments were raised in the report.
Circularity Check
No significant circularity
full rationale
The paper establishes universality of the log-asymptotics and entropic repulsion profile for stationary Gaussian fields with spectral singularity of order α by isolating the singular contribution |ξ|^{-α} near the origin via Fourier analysis, then reducing the large-deviation rate to the α-Riesz capacity and equilibrium potential under explicitly stated regularity conditions on the spectral measure. This generalizes the Bolthausen-Deuschel-Zeitouni GFF result without hidden Markov assumptions or self-citations that bear the central load; the derivation remains self-contained against external potential-theoretic benchmarks and does not reduce any claimed prediction or formula to a fitted input or prior result by definition.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Mild regularity conditions on the stationary Gaussian field with spectral singularity of order alpha
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
We compute the exact log-asymptotics of the persistence probability... explicit formulations in terms of the capacity and equilibrium potential of the α-Riesz kernel.
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IndisputableMonolith/Foundation/AlphaCoordinateFixation.leanJ_uniquely_calibrated_via_higher_derivative unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Cap_f(D) := min {∥h∥²_H : h∈H, h≥1 on D}
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Forward citations
Cited by 1 Pith paper
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Reference graph
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