Scaling limits of the Gaussian free field local-time pseudo-metric: the CLE$_4$ loop metric and Brownian-time identities on Riemann surfaces

On the metric graph (cable graph) of a finite weighted graph, the Gaussian free field (GFF) is a random continuous function generalizing one-dimensional Brownian bridges, and its zero set induces a natural pseudo-metric: the length of a path is the local time accumulated at level zero by the field along it. Lupu and Werner computed explicit laws for this pseudo-metric on finite graphs — for instance, the pseudo-distance between two boundary points depends only on the effective resistance between them — and formulated a program of continuum predictions: in two dimensions the pseudo-metric should converge, jointly with the outermost sign clusters, to the conformally invariant metric on CLE$_4$ loops (their Conjecture 15), and on general compact bordered Riemann surfaces the clusters of the critical Brownian loop soup should carry a corresponding metric together with Brownian-time distributional identities for GFF functionals (their Conjectures 16–18). A September 2026 construction by Kammerer, Kavvadias, Miller, and Tian of the conformally invariant CLE$_4$ loop metric — obtained as a renormalized $\kappa\downarrow 4$ limit of non-simple CLE graph metrics, realizing the previously unpublished Sheffield–Watson–Wu approach — settles the bare existence and uniqueness of the target metric, but not its identification with the GFF local-time pseudo-metric, nor any of the Riemann-surface predictions. The problem is to establish these scaling limits and identities.

Background

The Gaussian free field (GFF) is the centered Gaussian random generalized function associated with the Dirichlet energy $\sum c_{xy}(f(x)-f(y))^2$ of a weighted graph or a planar domain; on a discrete graph it is a random function on vertices. Given a finite weighted graph, its metric graph (or cable graph) replaces each edge by a continuous line segment of length equal to the edge resistance (the inverse of its conductance). Interpolating the discrete GFF on each edge by conditionally independent Brownian bridges yields the metric-graph GFF, a random continuous function on the graph. Because it is pointwise defined, one-dimensional Brownian techniques apply to it; in particular, its zero set is well defined. Lupu and Werner introduced a random pseudo-metric on the metric graph by declaring the length of a continuous path to be the cumulative local time at level zero (local time is the natural measure of the time a process spends at a point, normalized so that it is non-trivial in the diffusion limit) collected by the GFF along the path; the pseudo-distance $\delta_{x,A}$ between a point $x$ and a boundary set $A$ is the infimum of such lengths over paths joining them. On finite graphs they proved a pathwise transformation relating the metric-graph GFF to the reflected metric-graph GFF through this pseudo-distance — a generalization of Paul Lévy's theorem identifying the local time at zero of a Brownian motion with the supremum of another Brownian motion — and computed explicit laws: when the boundary consists of two points, the law of the pseudo-distance between them depends only on the effective resistance $R^{\mathrm{eff}}$ of the network between them (the total resistance of the graph viewed as an electric circuit), and more generally the pseudo-distance to a boundary set has an explicit Laplace transform expressed through Brownian bridges of length equal to effective resistances.

In two dimensions these discrete formulas are predicted to have conformally invariant continuum analogs. The conformal loop ensemble CLE$_4$ is a canonical, conformally invariant random countable collection of non-crossing loops in a simply connected planar domain, known to describe the scaling limit of the outermost sign-cluster boundaries of the metric-graph GFF. The CLE$_4$ couples to the continuum GFF in two ways reminiscent of the two classical constructions of Brownian motion from reflected Brownian motion: via independent random signs and the height gap (the fixed jump $2\lambda$ of level lines across loops), and via the conformally invariant growing mechanism of Werner and Wu known as the uniform exploration, which attaches to each loop $\Gamma$ a time parameter $t(\Gamma)>0$ (loops nearer the boundary are discovered earlier). Lupu and Werner conjectured that the metric-graph pseudo-distance converges to these time parameters, and noted that Sheffield, Watson and Wu had announced (in work that remained unpublished) that the times $t(\Gamma)$ extend to a full conformally invariant metric on the CLE$_4$ loops together with the domain boundary, the hard point being the symmetry $t_\Gamma(\Gamma')=t_{\Gamma'}(\Gamma)$ — a symmetry that is obvious on the metric graph. In September 2026, Kammerer, Kavvadias, Miller and Tian constructed this metric rigorously in three papers, as the renormalized limit as $\kappa\downarrow 4$ of the graph metric on the loops of a non-simple CLE$_\kappa$: it is conformally invariant, local, geodesic, uniquely characterized by these properties, and its metric-ball growth from the boundary is exactly the uniform exploration of Werner and Wu. However, their construction does not involve the GFF metric-graph pseudo-metric at all, so the identification of the two objects — the content of the GFF-side conjecture — remains open.

Beyond simply connected planar domains, Lupu and Werner formulated predictions for the continuum GFF on compact bordered Riemann surfaces (a surface with a complex structure whose boundary is a finite union of circles), where Schramm–Loewner evolution (SLE, the conformally invariant curve process describing scaling limits of planar interfaces) techniques largely break down and the notions of interior and exterior of a loop are ambiguous. There, the natural conformally invariant quantity replacing effective resistance is the extremal distance (also called extremal length), the conformal modulus of the curve family joining two boundary pieces, and the GFF is coupled to the critical Brownian loop soup (a Poissonian ensemble of Brownian loops at intensity $1/2$, corresponding to central charge $1$), whose outermost cluster boundaries are CLE$_4$ loops in the planar case. The continuum analogs of the first passage sets $\Lambda_a$ (the local set of points reachable from the boundary by a path on which the field stays above level $a$, rigorously constructed in finitely connected planar domains by Aru, Lupu and Sepúlveda) are the objects through which the predicted identities are stated. The simply connected planar prototype of one of these identities — the log-conformal-radius formula, in which a log-ratio of conformal radii of the first passage set is distributed as a Brownian hitting time — is already a theorem, which is part of the evidence for the surface-level predictions.

Problem Statement

Establish the two-dimensional continuum theory of the Gaussian-free-field local-time pseudo-metric predicted by Lupu and Werner, in the following precise, jointly linked components (the common objective is to show that the explicit finite-graph laws for the pseudo-metric have conformally invariant continuum counterparts, centered on one and the same object — the metric on CLE$_4$ / critical loop-soup clusters):

1. (Conjecture 15 of the source.) Let $D$ be an open bounded simply connected planar domain, $D_\varepsilon$ a fine-mesh approximation of $D$, $A_\varepsilon$ the approximation of $\partial D$, and $\tilde\varphi_\varepsilon$ the metric-graph GFF on $D_\varepsilon$ with zero boundary condition on $A_\varepsilon$. Prove that, as $\varepsilon\to 0$, the local-time pseudo-distance $\delta_{x,A}(\tilde\varphi_\varepsilon)$ converges — jointly with the outermost sign clusters of $\tilde\varphi_\varepsilon$ — to the time parameters $t(\Gamma)$ that the conformally invariant uniform exploration of Werner and Wu attaches to the loops $\Gamma$ of the CLE$_4$ coupled to the continuum GFF. In particular, the scaling limit of the pseudo-metric should be the conformally invariant metric on CLE$_4$ loops now constructed by Kammerer–Kavvadias–Miller–Tian (which realizes the previously unpublished Sheffield–Watson–Wu construction); proving this would identify the two constructions and, as the source notes, yield an alternative, GFF-side proof of the existence and symmetry of that metric.

2. (Conjecture 16 of the source.) Let $\Sigma$ be a compact bordered Riemann surface whose border is partitioned into three parts $B_1(\Sigma),B_2(\Sigma),B_3(\Sigma)$ with $B_1,B_2$ non-empty and at positive distance (each part consisting of finitely many connected components, and $B_3(\Sigma)$ allowed to be empty), and let $\mathcal L_{g,1/2}$ be the critical Brownian loop soup (intensity $1/2$) on $\Sigma$, built from Brownian motion on $\Sigma$ killed on the border $B_1(\Sigma)\cup B_2(\Sigma)$ and instantaneously normally reflected on $B_3(\Sigma)$. Prove that there is a metric on the clusters of $\mathcal L_{g,1/2}$ inside $\Sigma$, measurable with respect to the loop soup, obtained as the limit of the metric-graph pseudo-metrics on fine-mesh approximations of $\Sigma$, and independent of the time-parametrization of the loops and of the choice of conformal metric $g$ compatible with the complex structure of $\Sigma$; that in the case where $\Sigma$ is a simply connected planar domain this metric is the conformally invariant CLE$_4$ metric of component 1; and that, as in the finite-graph Laplace-transform formulas, the square of the distance between $B_1(\Sigma)$ and $B_2(\Sigma)$ induced by this metric is an exponential random variable with mean $2\,\mathrm{ED}(B_1(\Sigma),B_2(\Sigma))$, twice the extremal distance between the two boundary parts.

3. (Conjecture 17 of the source.) In the same surface setting, with the GFF $\varphi^\Sigma$ having constant boundary condition $h_2$ on $B_2(\Sigma)$, continuous bounded boundary data $h_1$ on $B_1(\Sigma)$ with $a<\min_{B_1(\Sigma)} h_1$, free (Neumann) condition on $B_3(\Sigma)$, and $m_{\mathrm{exc}}$ the mean value of $h_1$ at the starting point of an excursion under the normalized $B_1$-to-$B_2$ Brownian excursion measure (the continuum counterpart of the finite-graph quantity denoted $m$ in the source's Proposition 13), first construct the level-$a$ first passage local set $\Lambda^\Sigma_a$ of $\varphi^\Sigma$ — the minimal compact local set containing $B_1(\Sigma)$, measurable in $\varphi^\Sigma$, such that conditionally on it the restricted field is a GFF with boundary value $a$ on $\Lambda^\Sigma_a$, $h_2$ on $B_2(\Sigma)\setminus\Lambda^\Sigma_a$ and free on $B_3(\Sigma)\setminus\Lambda^\Sigma_a$ (the construction is part of the problem) — and then prove that the extremal distance $\mathrm{ED}(\Lambda^\Sigma_a,B_2(\Sigma))$ is distributed as the last visit time of the level $a$ by a Brownian bridge of length $\mathrm{ED}(B_1(\Sigma),B_2(\Sigma))$ from $h_2$ to $m_{\mathrm{exc}}$ (defined as $0$ if the bridge never reaches $a$).

4. (Conjecture 18 of the source.) With $\Lambda^\Sigma_a(z_0)$, for an interior point $z_0\in\Sigma\setminus\partial\Sigma$, the level-$a$ first passage local set containing $B_1(\Sigma)\cup B_2(\Sigma)$ — the minimal compact local set of $\varphi^\Sigma$, measurable in $\varphi^\Sigma$, such that $\Sigma\setminus\Lambda^\Sigma_a(z_0)$ is connected and contains $z_0$, and conditionally on it the restricted field is a GFF with boundary value $a$ on $\Lambda^\Sigma_a(z_0)$ and free on $B_3(\Sigma)\setminus\Lambda^\Sigma_a(z_0)$ — and with $G_\Sigma$ the Green function of the Laplacian with zero Dirichlet condition on $B_1\cup B_2$ and zero Neumann condition on $B_3$, prove that $G_\Sigma(z_0,z_0)-G_{\Sigma\setminus\Lambda^\Sigma_a(z_0)}(z_0,z_0)$ is distributed as the first hitting time of the level $a$ by a standard Brownian motion started at $m$, the value at $z_0$ of the harmonic extension of the boundary data $h$ (with zero Neumann condition on $B_3(\Sigma)$).

All four components are mandatory and each must be proved in its own right. Components 1 and 2 are two faces of one question (the continuum metric and its two constructions: from the GFF local-time pseudo-metric and from the CLE$_4$/loop-soup side); components 3 and 4 are the distributional identities that the same finite-graph Brownian-bridge formulas predict for the first passage sets of the continuum GFF, and while they are expected to follow from the same scaling-limit mechanism, that expectation is not a substitute for their proof.

References

1. Titus Lupu, Wendelin Werner. The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph. Probability Theory and Related Fields 171 (2018) 775–818. DOI: 10.1007/s00440-017-0792-y; arXiv:1607.06424. https://doi.org/10.1007/s00440-017-0792-y
2. Emmanuel Kammerer, Konstantinos Kavvadias, Jason Miller, Yi Tian. The conformally invariant metric on CLE$_4$ I: subsequential limits of the non-simple CLE graph metric. arXiv:2609.04138 (2026). https://arxiv.org/abs/2609.04138
3. Emmanuel Kammerer, Konstantinos Kavvadias, Jason Miller, Yi Tian. The conformally invariant metric on CLE$_4$ II: existence of geodesics. arXiv:2609.04139 (2026). https://arxiv.org/abs/2609.04139
4. Emmanuel Kammerer, Konstantinos Kavvadias, Jason Miller, Yi Tian. The conformally invariant metric on CLE$_4$ III: uniqueness. arXiv:2609.04140 (2026). https://arxiv.org/abs/2609.04140
5. Wendelin Werner, Hao Wu. On conformally invariant CLE explorations. Communications in Mathematical Physics 320 (2013) 637–661. DOI: 10.1007/s00220-013-1719-9. https://doi.org/10.1007/s00220-013-1719-9
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9. Juhan Aru, Titus Lupu, Avelio Sepúlveda. Extremal distance and conformal radius of a CLE$_4$ loop. Annals of Probability 50 (2022) 509–558. DOI: 10.1214/21-AOP1538; arXiv:2004.13570. https://doi.org/10.1214/21-AOP1538
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12. Titus Lupu. Convergence of the two-dimensional random walk loop-soup clusters to CLE. Journal of the European Mathematical Society 21 (2019) 1201–1227. DOI: 10.4171/jems/859; arXiv:1502.06827. https://doi.org/10.4171/jems/859