Bounded-type thin local sets of the 2D Gaussian free field: maximality of nested CLE_4 carpets and removal of the topological regularity condition

A bounded-type thin local set (BTLS) of the two-dimensional Gaussian free field (GFF) is a local set whose associated mean-value (harmonic) function is deterministically bounded and which carries no GFF mass on itself. Aru, Sepúlveda and Werner introduced this class and proved that every BTLS is contained in a nested CLE_4 carpet: a $(2M\lambda)$-BTLS lies in the $(M+1)$-fold nested carpet $\mathrm{CLE}_4^{M+1}$. They noted two limitations of the theory that they left open: (1) sharpness of the containment -- is every $(2M\lambda)$-BTLS already contained in $\mathrm{CLE}_4^M$, which would make $\mathrm{CLE}_4^M$ the maximal element of the class and complete the analogy with exit times of intervals for one-dimensional Brownian motion; and (2) robustness of the definition -- do the main classification statements survive if the auxiliary topological regularity condition in the BTLS definition (locally separated interior connected components) is dropped, with the harmless normalization that $A$ has no isolated points retained, as the authors conjectured without proof. A third question raised in the same paper -- determining precisely when the height labels of the two-valued sets $A_{-a,b}$ are a measurable function of the set -- has since been resolved by Aru and Sepúlveda (2018) and is not part of this problem.

Background

The Gaussian free field (GFF) in a planar domain $D$ is a centered Gaussian random generalized function, the natural analogue of Brownian motion when the time axis is replaced by a two-dimensional domain; like Brownian motion it is not defined pointwise, but it has a rich Markovian structure. A random closed set $A$ coupled with a GFF $\Gamma$ is called a local set if, conditionally on $A$, the field in $D\setminus A$ decomposes as a conditionally independent zero-boundary GFF plus a random harmonic function $h_A$; this is the generalization of a stopping time to multidimensional time, with $h_A$ interpreted as the harmonic extension of the values of the field on $\partial A$. The set $A$ is thin if it carries no GFF mass on itself, meaning that for every smooth test function $f$ the field integrates to $\int_{D\setminus A}h_A(x)f(x)\,dx$ plus an independent GFF contribution; and it is of bounded type if $|h_A|\le K$ almost surely for some deterministic $K$ (a $K$-BTLS). Boundedness is a strong requirement: informally it asks the GFF itself to be bounded by $K$ on $\partial A$, and it forces $A$ to be random and small (its Minkowski, i.e. box-counting, dimension is bounded by a constant $d(K)<2$ depending on $K$). In the normalization of the source paper the constant $\lambda=\sqrt{\pi/8}$ plays the role of the height gap: level lines of the GFF at heights $\pm\lambda$, first constructed on the lattice and then in the continuum by Schramm and Sheffield, are SLE$_4$ curves (the Schramm-Loewner evolution with parameter $4$, a random conformally invariant simple curve), and the outermost loops of the conformal loop ensemble $\mathrm{CLE}_4$ (the loop ensemble describing scaling limits of clusters in critical models, constructed by Sheffield and Werner) coupled with the GFF form a $(2\lambda)$-BTLS with $h_A\in\{-2\lambda,2\lambda\}$. Iterating this exploration $M$ times produces the nested carpet $\mathrm{CLE}_4^M$, a $(2M\lambda)$-BTLS whose harmonic function equals $\pm 2M\lambda$ in each component of its complement; it is constructed from the label process, which seen from any fixed point evolves like a simple random walk in units of $2\lambda$. Aru, Sepúlveda and Werner proved the basic classification of this paper's title: a local set is a BTLS if and only if it is contained in some nested $\mathrm{CLE}_4^M$ carpet; more precisely every $(2M\lambda)$-BTLS is almost surely contained in $\mathrm{CLE}_4^{M+1}$ (their Proposition 3); every BTLS satisfies that $A\cup\partial D$ is connected; and for $-a<0<b$ with $a+b\ge 2\lambda$ there exists a unique two-valued BTLS $A_{-a,b}$ with $h_A\in\{-a,b\}$, measurable as a function of the GFF and monotone in $[a,b]$ (Proposition 2), while $\mathrm{CLE}_4^M$ is the unique BTLS with values in $\{-2M\lambda,2M\lambda\}$ (Proposition 1). The conceptual picture is that BTLS are the GFF analogue of bounded stopping times of one-dimensional Brownian motion and the nested carpets are the analogues of exit times of intervals: a stopping time with $|B_T|\le K$ is dominated by the exit time from $[-K,K]$. Two features of the actual results are in tension with this analogy. First, the containment loses one layer of nesting: the proof only yields $\mathrm{CLE}_4^{M+1}$ for a $(2M\lambda)$-BTLS, and the authors state that the expected sharper conclusion seems to require additional technical work. Second, the BTLS definition contains an auxiliary topological condition: almost surely $A$ has no isolated points, and every connected component of $A$ that does not meet $\partial D$ has a neighbourhood that meets no other component of $A$ (which implies there are at most countably many interior components). This condition excludes, for example, totally disconnected sets that are non-polar for Brownian motion -- that is, hit by Brownian motion with positive probability -- such as Cantor sets of Hausdorff dimension in $(0,1)$; it is used at several load-bearing points of the proofs, and the authors believe it is dispensable, i.e., that all their statements would hold without it, but did not prove this. A general introduction to the GFF and its interaction with $\mathrm{CLE}_4$ can be found in the lecture notes of Werner and Powell.

Problem Statement

Both questions below serve one objective: a sharp, hypothesis-free version of the correspondence between bounded-type thin local sets of the GFF and nested CLE$_4$ carpets. Throughout, $D$ is a simply connected planar domain, $\Gamma$ is a zero-boundary GFF in $D$, $\lambda=\sqrt{\pi/8}$, 'BTLS' means the class of Definition 7 of Aru-Sepúlveda-Werner (a thin local set with $|h_A|\le K$ almost surely for some deterministic $K$, satisfying the topological condition (3')), and $\mathrm{CLE}_4^M$ denotes the nested carpet coupled with $\Gamma$ constructed there -- the $(2M\lambda)$-BTLS whose harmonic function takes the values $\pm 2M\lambda$ in the components of its complement, itself a measurable function of $\Gamma$. (a) Maximality. Prove or disprove: for every integer $M\ge 1$, every BTLS $B$ coupled with $\Gamma$ such that $|h_B|\le 2M\lambda$ almost surely (in the terminology of the source, a $(2M\lambda)$-BTLS) is almost surely contained in $\mathrm{CLE}_4^M$. The source proves only $B\subseteq\mathrm{CLE}_4^{M+1}$ (Proposition 3); a positive answer makes $\mathrm{CLE}_4^M$ the maximal element of the class of $(2M\lambda)$-BTLS and completes the analogy with exit times of intervals for bounded stopping times of one-dimensional Brownian motion, while a negative answer requires an explicit $(2M\lambda)$-BTLS not contained in $\mathrm{CLE}_4^M$. (b) Definitional robustness. Prove or disprove: the main classification statements of the source -- (i) a local set is a BTLS if and only if it is contained in some nested $\mathrm{CLE}_4^M$ carpet; (ii) $A\cup\partial D$ is connected for every BTLS; (iii) the existence, uniqueness, monotonicity and GFF-measurability of the two-valued sets $A_{-a,b}$ for $-a<0<b$ with $a+b\ge 2\lambda$; and (iv) the uniqueness of $\mathrm{CLE}_4^M$ among BTLS with $h_A\in\{-2M\lambda,2M\lambda\}$ -- remain valid when the component-separation requirement of the topological condition (3') (each connected component of $A$ not meeting $\partial D$ has a neighbourhood meeting no other component) is deleted from the BTLS definition, i.e., for local, thin sets with deterministically bounded $h_A$ that have no isolated points (the harmless normalization) and with no separation assumption on the connected components. These four statements are the main classification content of the source; the authors' dispensability belief is stated for all results of the paper, and the asked problem targets precisely these four. The authors believe all statements survive without the condition but provide no proof; a negative answer requires exhibiting a bounded-type thin local set without the topological condition for which one of the named statements fails. A third question raised in the same paper -- determining precisely when the labels of $A_{-a,b}$ are a measurable function of the set -- was answered by Aru and Sepúlveda (2018) and is not part of this problem.

References

1. Juhan Aru, Avelio Sepúlveda, Wendelin Werner. On bounded-type thin local sets of the two-dimensional Gaussian free field. Journal of the Institute of Mathematics of Jussieu 18(3):591-618 (2019; online 2017). DOI: 10.1017/s1474748017000160; arXiv:1603.03362. https://doi.org/10.1017/s1474748017000160
2. Juhan Aru, Avelio Sepúlveda. Two-valued local sets of the 2D continuum Gaussian free field: connectivity, labels, and induced metrics. Electronic Journal of Probability 23, Paper No. 61, 35 pp. (2018). DOI: 10.1214/18-EJP182; arXiv:1801.03828. https://doi.org/10.1214/18-EJP182
3. Juhan Aru, Titus Lupu, Avelio Sepúlveda. First passage sets of the 2D continuum Gaussian free field. Probability Theory and Related Fields 176(3-4):1303-1355 (2020; online 2019). DOI: 10.1007/s00440-019-00941-1; arXiv:1706.07737. https://doi.org/10.1007/s00440-019-00941-1
4. Avelio Sepúlveda. On thin local sets of the Gaussian free field. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 55(3):1797-1813 (2019). DOI: 10.1214/19-AIHP1005; arXiv:1702.03164. https://doi.org/10.1214/19-AIHP1005
5. Lukas Schoug, Avelio Sepúlveda, Fredrik Viklund. Dimensions of Two-Valued Sets via Imaginary Chaos. International Mathematics Research Notices 2022(5):3219-3261 (online 2020). DOI: 10.1093/imrn/rnaa250; arXiv:1910.09294 (preprint title: 'Dimension of two-valued sets via imaginary chaos'). https://doi.org/10.1093/imrn/rnaa250
6. Juhan Aru, Titus Lupu, Avelio Sepúlveda. The first passage sets of the 2D Gaussian free field: convergence and isomorphisms. Communications in Mathematical Physics 375(3):1885-1929 (2020). DOI: 10.1007/s00220-020-03718-z; arXiv:1805.09204. https://doi.org/10.1007/s00220-020-03718-z
7. Juhan Aru, Titus Lupu, Avelio Sepúlveda. Extremal distance and conformal radius of a CLE4 loop. The Annals of Probability 50(2) (2022). DOI: 10.1214/21-AOP1538; arXiv:2004.13570 (preprint title: 'Extremal distance and conformal radius of a CLE_4 loop'). https://doi.org/10.1214/21-AOP1538
8. Juhan Aru, Titus Lupu, Avelio Sepúlveda. Excursion decomposition of the 2D continuum GFF. arXiv:2304.03150 (2023). https://arxiv.org/abs/2304.03150
9. Wendelin Werner, Ellen Powell. Lecture notes on the Gaussian Free Field. arXiv:2004.04720 (2020). https://arxiv.org/abs/2004.04720
10. Oded Schramm, Scott Sheffield. A contour line of the continuum Gaussian free field. Probability Theory and Related Fields 157(1-2):47-80 (2013; online 2012). DOI: 10.1007/s00440-012-0449-9; arXiv:1008.2447. https://doi.org/10.1007/s00440-012-0449-9
11. Oded Schramm, Scott Sheffield. Contour lines of the two-dimensional discrete Gaussian free field. Acta Mathematica 202(1):21-137 (2009). DOI: 10.1007/s11511-009-0034-y. https://doi.org/10.1007/s11511-009-0034-y
12. Scott Sheffield, Wendelin Werner. Conformal loop ensembles: The Markovian characterization and the loop-soup construction. Annals of Mathematics 176(3):1827-1917 (2012). DOI: 10.4007/annals.2012.176.3.8; arXiv:1006.2374. https://doi.org/10.4007/annals.2012.176.3.8