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arxiv: 2606.11043 · v1 · pith:SPB735MSnew · submitted 2026-06-09 · 🧮 math.PR

Thick points under Gaussian free field dynamics

Felipe Espinosa Vergara , Avelio Sep\'ulveda This is my paper

Pith reviewed 2026-06-27 11:52 UTC · model grok-4.3

classification 🧮 math.PR
keywords thick pointsGaussian free fieldOrnstein-Uhlenbeckstochastic heat equationphase transitionsBeraha numbersFK percolationexceptional times
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The pith

Thickness of points in the 2D Gaussian free field evolves continuously under the Ornstein-Uhlenbeck dynamic for all points at once, while the additive stochastic heat equation produces discontinuous thickness, super-thick points, and phase

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

The paper examines how the thickness of points in the two-dimensional Gaussian free field changes over time under two different dynamics. For the Ornstein-Uhlenbeck dynamic, it shows that thickness changes continuously everywhere at once and identifies exactly which functions can arise as thickness paths for some points. For the stationary additive stochastic heat equation, thickness can jump and points can become thicker than the usual maximum of two, leading to infinitely many critical values of the parameter gamma where the number of very thick points changes. These critical values map to the Beraha numbers in the FK percolation model. A reader would care because these results link the local behavior of the random field to global phase transitions in related statistical mechanics models.

Core claim

Under the Ornstein-Uhlenbeck dynamics for the 2D GFF, the thickness of every point evolves continuously in time, and the possible thickness functions are all deterministic functions f from reals to reals. Under the stationary solution of the additive stochastic heat equation, thickness is discontinuous and points with thickness exceeding 2 exist; moreover, for each N there are phase transitions in gamma >2 at specific values gamma squared equals 8, 6, 16/3 and so on where the existence of times with at least N gamma-thick points changes, and these values are the Beraha numbers via the map q equals 4 cos squared of 4 pi over gamma squared.

What carries the argument

The thickness of a point, extracted from the local covariance structure of the GFF, tracked simultaneously under the Ornstein-Uhlenbeck process and under the stationary additive stochastic heat equation.

If this is right

  • Every deterministic real-valued function on the time line arises as the thickness trajectory of some point under the Ornstein-Uhlenbeck dynamic.
  • The additive stochastic heat equation produces points whose thickness exceeds 2.
  • For each fixed N there exist infinitely many critical gamma greater than 2 at which the existence of times with at least N gamma-thick points changes.
  • These critical gamma squared values are 8, 6, 16/3 and so on, converge to 4, and map exactly to the Beraha numbers under the FK-model relation q equals 4 cos squared of 4 pi over gamma squared.
  • At these Beraha values the associated conformal field theory for critical FK percolation is minimal.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The phase transitions supply a dynamical mechanism that could realize the Beraha-number spectrum inside a single random field rather than across a family of static models.
  • The convergence of the critical values to 4 as N grows suggests a limiting regime in which the number of exceptionally thick points becomes dense at the boundary gamma squared equals 4.
  • The same continuity-versus-discontinuity distinction may appear when thickness is replaced by other local functionals such as the Liouville measure or the occupation measure of the field.
  • Because the mapping to FK percolation is explicit, the exceptional times identified here correspond to dynamical versions of the points where the FK model changes its conformal properties.

Load-bearing premise

The thickness measure is well-defined from the GFF covariance and the two dynamics preserve the properties needed for the continuity and phase-transition statements to hold.

What would settle it

Observation of a single point whose thickness function jumps under the Ornstein-Uhlenbeck dynamics, or failure to detect a change in the number of gamma-thick points when gamma squared crosses one of the listed values such as 8 or 6.

Figures

Figures reproduced from arXiv: 2606.11043 by Avelio Sep\'ulveda, Felipe Espinosa Vergara.

Figure 1
Figure 1. Figure 1: Simulation of the evolution of the thickness of the point x that attains the maximum at time 0 of the OU-GFF. The red lines indicate the value 1/ p log(N) which is the typical thickness of points in this simulation, where N = 2000. For videos of these simulations, see [Simb]. The main results of this article concern the behavior of thick points in each of the two dynamics. For the OU-GFF, we study the thic… view at source ↗
Figure 2
Figure 2. Figure 2: Simulation of the evolution of the maximum thickness of the SHEF. The red lines indicate the value 2− 3 4 log(log(N)) log(N) which is the typical value for the maximum of a GFF in the grid N × N, here N = 1000. Note that as the size is small, the log-log correction is actually macroscopic. For videos of these simulations, see [Sima]. where (αk)k∈N is an i.i.d. sequence of standard Gaussian random variables… view at source ↗
Figure 3
Figure 3. Figure 3: Hausdorff dimension of T γ as a function of γ 2 . The separation of the regimes “thick” and “super-thick” is motivated by 2 observations. The first one is that since T γ has non-trivial Hausdorff dimension, there have to exist random times with super-thick points which implies that at those times the model is not absolutely continuous with respect to the GFF. The second one is that, one can interpret the l… view at source ↗
read the original abstract

We investigate the evolution of thick points under two natural dynamics for the Gaussian free field (GFF) in dimension 2. The first dynamic we analyze is the Ornstein-Uhlenbeck GFF. We prove that, simultaneously for all points, the evolution of their thickness is continuous. Additionally, we characterize all deterministic functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that there are points whose thickness function is $f$. The second dynamic we study is the stationary solution of the additive stochastic heat equation. In this case, the thickness of points is not continuous. Moreover, this rougher dynamic generates super-thick points, namely points with thickness greater than $2$. As a function of $\gamma > 2$, we identify infinitely many phase transitions corresponding to the existence of exceptional times where at least $N$ points are $\gamma$-thick. These phase transitions, occurring at $\gamma^2 = 8, 6, 16/3, \dots$, converge to $4$ as $N \to \infty$. Mapping to the critical FK-model parameter via $q=4\cos^2(4\pi/\gamma^2)$, these critical values correspond to the Beraha numbers, which are precisely the points at which the CFT for critical FK percolation should be minimal

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

0 major / 3 minor

Summary. The manuscript investigates the evolution of thick points of the 2D Gaussian free field under two dynamics: the Ornstein-Uhlenbeck GFF and the stationary additive stochastic heat equation. For the OU dynamics it claims simultaneous continuity of thickness for all points and a complete characterization of admissible deterministic thickness functions f. For the SHE dynamics it claims discontinuity of thickness, the existence of super-thick points (thickness >2), and the location of infinitely many phase transitions in gamma>2 for the existence of at least N gamma-thick points at the values gamma^2=8,6,16/3,..., converging to 4; these are mapped to Beraha numbers via q=4 cos^2(4 pi / gamma^2) and linked to minimal CFT for critical FK percolation.

Significance. If the stated continuity, characterization, and phase-transition results hold with the indicated rigor, the work supplies a precise dynamical analysis of exceptional sets for log-correlated fields, connecting the geometry of thick points to stochastic PDEs and to the Beraha numbers of FK percolation. The explicit sequence of critical gamma values and the observation that they accumulate at the Liouville quantum gravity threshold constitute a concrete, falsifiable contribution to the interface between Gaussian multiplicative chaos and conformal field theory.

minor comments (3)
  1. The definition of thickness (via cutoffs or limits on the circle-average or other regularization of the GFF) should be stated explicitly in §1 or §2 before the statements of the main theorems, even if it follows standard references; this would make the continuity claim in the OU case immediately verifiable from the text.
  2. In the phase-transition statement, the precise meaning of 'exceptional times' (Lebesgue measure zero but positive Hausdorff dimension, or existence in the support of the measure) should be clarified in the paragraph following the list of gamma^2 values.
  3. The mapping q=4 cos^2(4 pi / gamma^2) is presented as an observation; a short remark confirming that the critical values indeed land exactly on the Beraha sequence (rather than approximately) would strengthen the CFT link without altering the probabilistic content.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the recognition of its significance at the interface of Gaussian multiplicative chaos, stochastic PDEs, and conformal field theory, and the recommendation of minor revision. No specific major comments or concerns were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives its main results on continuity of thickness under OU dynamics, characterization of admissible thickness functions, non-continuity and super-thick points under additive SHE, and the explicit locations of gamma-phase transitions directly from the standard log-correlated covariance structure of the 2D GFF together with explicit constructions and analysis of the two dynamics. The Beraha-number mapping is presented strictly as a post-hoc observation after the transition values are obtained, not as an input or justification. No self-definitional reductions, fitted quantities renamed as predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the ledger is populated with the minimal domain assumptions required to state the problem.

axioms (1)
  • domain assumption The two-dimensional Gaussian free field exists with its standard logarithmic covariance kernel.
    The entire study is built on this object; invoked implicitly throughout the abstract.

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