Metric-Graph Gaussian Free Field

• Metric-Graph GFF is a continuous Gaussian free field defined on cable systems where each edge is replaced by a line segment with Brownian bridge interpolation.
• It preserves the Green function covariance structure and exhibits a strong Markov property, enabling precise definitions of level sets and intrinsic pseudo-metrics.
• By linking discrete GFFs with continuum models, it offers deep insights into planar percolation, loop soups, and high-dimensional critical phenomena.

Searching arXiv for recent and foundational papers on metric-graph Gaussian free fields and related level-set geometry. The metric-graph Gaussian free field (metric-graph GFF), also called the cable-system GFF, is the Gaussian free field on a one-dimensional network obtained by replacing each edge of a weighted graph by a line segment and endowing the resulting space with its natural path metric. It is a continuous Gaussian field whose restriction to the vertex set is the usual discrete GFF, while along each edge it is given by a Brownian-bridge interpolation between endpoint values. This construction makes the field pointwise defined and continuous on the cable system, preserves the Green-function covariance structure, and provides a natural bridge between discrete lattice models, Brownian loop soups, first passage sets, and continuum Gaussian free fields (Bi et al., 26 Mar 2026).

1. Construction on cable systems

Let $\mathcal G=(V,E)$ be a finite connected graph with positive conductances $C(x,y)$. The associated metric graph $\widetilde{\mathcal G}$ is obtained by replacing each edge $\{x,y\}$ by a line segment of length $C(x,y)^{-1}$, i.e. its resistance; in the two-dimensional box setup used for one-arm probabilities on $\mathbb Z^2$, each nearest-neighbor edge is replaced by a segment of length $2$ (Aru et al., 2018, Lupu et al., 2016, Bi et al., 26 Mar 2026). Brownian motion on $\widetilde{\mathcal G}$ behaves as one-dimensional Brownian motion along edges and, at vertices, continues along adjacent edges according to the underlying graph structure; with equal conductances this continuation is uniform among outgoing edges, and more generally it is weighted by the conductances (Lupu et al., 2016).

The metric-graph GFF $\tilde\phi$ with Dirichlet boundary condition is the centered Gaussian process indexed by points of $\widetilde{\mathcal G}$ whose covariance is the Green function of this Brownian motion killed on the boundary. Equivalently, one first samples the discrete GFF $C(x,y)$0 on $C(x,y)$1, then, conditionally on the vertex values, fills each edge independently with a Brownian bridge joining the endpoint values. Thus $C(x,y)$2 is the unique continuous Gaussian field whose vertex restriction is $C(x,y)$3, whose conditional mean along edges is harmonic, and whose covariance matches the cable-system Green function (Aru et al., 2018, Lupu et al., 2016).

For the standard two-dimensional box

$C(x,y)$4

the discrete field with Dirichlet boundary on the inner boundary of $C(x,y)$5 has covariance given by the killed simple-random-walk Green function $C(x,y)$6, and the metric-graph field $C(x,y)$7 on $C(x,y)$8 is its cable-system extension with

$C(x,y)$9

Restricted to vertices, $\widetilde{\mathcal G}$0 is exactly the discrete GFF $\widetilde{\mathcal G}$1 (Bi et al., 26 Mar 2026).

2. Markovian structure, first passage sets, and intrinsic geometry

A decisive advantage of the cable-system setting is that the GFF is a continuous random function on a one-dimensional network and satisfies a strong Markov property along random compact sets. If $\widetilde{\mathcal G}$2 is an optional random compact set with finitely many connected components, then

$\widetilde{\mathcal G}$3

where, conditionally on $\widetilde{\mathcal G}$4, the field $\widetilde{\mathcal G}$5 is an independent zero-boundary metric-graph GFF on $\widetilde{\mathcal G}$6, and $\widetilde{\mathcal G}$7 is harmonic off $\widetilde{\mathcal G}$8 with boundary values prescribed on $\widetilde{\mathcal G}$9 and on the original boundary (Aru et al., 2018). This is the cable-system analogue of the local-set decomposition for the continuum GFF.

This structure permits a literal pathwise definition of first passage sets. For a harmonic boundary condition $\{x,y\}$0 and a real parameter $\{x,y\}$1, the first passage set of level $\{x,y\}$2 is

$\{x,y\}$3

Every connected component of $\{x,y\}$4 touches the boundary; on the new boundary created by the set, the field equals $\{x,y\}$5; and the complement carries an independent zero-boundary metric-graph GFF. These sets are monotone in both the level and the boundary condition (Aru et al., 2018). In the metric-graph setting this pathwise description is not heuristic: it is the definition.

The cable-system formalism also supports an intrinsic pseudo-metric built from the zero set. For a continuous path $\{x,y\}$6, its length is defined as the local time at level $\{x,y\}$7 accumulated by $\{x,y\}$8, and the induced pseudo-distance is

$\{x,y\}$9

This pseudo-distance vanishes exactly on sign clusters, since points connected without crossing the zero set accumulate zero local time. A Lévy-type identity relates the pair $C(x,y)^{-1}$0 to a reflected/unreflected transformation of the field, extending classical identities for Brownian motion and Brownian bridges to general metric graphs (Lupu et al., 2016). When the boundary consists of two points, the law of the pseudo-distance depends only on the effective resistance between them and coincides with the law of the local time at zero of a one-dimensional Brownian bridge of corresponding length (Lupu et al., 2016).

3. Planar level-set percolation and one-arm probabilities

For a level $C(x,y)^{-1}$1, the metric-graph excursion set is

$C(x,y)^{-1}$2

with connectivity defined by continuous paths in the cable system. In the discrete model, the corresponding set is

$C(x,y)^{-1}$3

with nearest-neighbor connectivity. The central observables are one-arm events such as

$C(x,y)^{-1}$4

and their discrete counterparts (Bi et al., 26 Mar 2026).

In two dimensions, the Green function satisfies $C(x,y)^{-1}$5, and the metric-graph bulk one-arm probability reflects this logarithmic variance scale. For any fixed $C(x,y)^{-1}$6,

$C(x,y)^{-1}$7

More concretely, for fixed $C(x,y)^{-1}$8 and large $C(x,y)^{-1}$9, this probability is essentially of order $\mathbb Z^2$0, whereas for $\mathbb Z^2$1 it lies between $\mathbb Z^2$2 and $\mathbb Z^2$3 for suitable constants $\mathbb Z^2$4 (Bi et al., 26 Mar 2026). The metric-graph model also admits conditional one-arm asymptotics given the field value at the origin, expressed through an exploration martingale and Brownian hitting probabilities (Bi et al., 26 Mar 2026).

The point-to-boundary regime exhibits a phase transition at level $\mathbb Z^2$5. For $\mathbb Z^2$6, Lupu–Werner’s isomorphism yields the exact identity

$\mathbb Z^2$7

hence

$\mathbb Z^2$8

For $\mathbb Z^2$9, by contrast,

$2$0

for some $2$1, so the decay is polynomial in $2$2 rather than logarithmic (Bi et al., 26 Mar 2026). This sharp distinction is specific to the boundary one-arm event.

A structural reason for the tractability of the cable model is that in a metric-graph exploration of $2$3, the explored boundary is exactly at level $2$4. The discrete model lacks this flat terminal condition: when the discrete exploration stops, its outer neighbors are below $2$5, but not pinned to $2$6. This discrepancy is the source of later quantitative differences between the two models (Bi et al., 26 Mar 2026).

4. Comparison with the discrete GFF and planar scaling limits

The discrete two-dimensional GFF has the same logarithmic bulk one-arm order as the cable model, but the probabilities are not asymptotically equal. For fixed $2$7,

$2$8

for all large $2$9, with $\widetilde{\mathcal G}$0 (Bi et al., 26 Mar 2026). Thus the discrete one-arm probability is strictly larger by an amount of the same order as the probability itself. In the discrete exploration, boundary values are typically strictly below $\widetilde{\mathcal G}$1, and the resulting entropic repulsion shifts the connection probability upward relative to the metric-graph case (Bi et al., 26 Mar 2026).

This discrete-versus-metric distinction is visible already in planar crossing events. On a rectangle $\widetilde{\mathcal G}$2 with zero boundary, the metric-graph probability of a positive crossing between inner boundary arcs is at most $\widetilde{\mathcal G}$3, while the corresponding discrete probability stays bounded away from both $\widetilde{\mathcal G}$4 and $\widetilde{\mathcal G}$5 (Ding et al., 2020). With alternating boundary data, both models have nondegenerate left-right crossing probabilities, but the discrete crossing probability exceeds the metric-graph one by a fixed positive amount for all sufficiently small mesh (Ding et al., 2020).

The scaling limit of the cable model with alternating boundary data is explicitly computable. On polygons of $\widetilde{\mathcal G}$6 with alternating $\widetilde{\mathcal G}$7 boundary values, the connection pattern of the frontiers of positive and negative first passage sets converges to a conformally covariant limit expressed through fused multiple $\widetilde{\mathcal G}$8 partition functions (Liu et al., 2020). In the rectangle case, if $\widetilde{\mathcal G}$9 is the conformal cross-ratio, then

$\tilde\phi$0

whereas the corresponding discrete GFF limit with the critical boundary amplitude is $\tilde\phi$1 (Liu et al., 2020, Ding et al., 2020). The fact that both models converge to the same continuum GFF but have different level-set crossing limits is one of the clearest indications that cable interpolation changes the geometry of excursion connectivity.

Metric-graph first passage sets also converge to continuum first passage sets in the Hausdorff metric, and certain natural metric-graph interfaces converge to continuum level lines and hence to $\tilde\phi$2 processes (Aru et al., 2018). More recently, continuum GFF level-line crossing probabilities with piecewise constant Dirichlet data were identified as ratios of $\tilde\phi$3 degenerate conformal blocks, and the corresponding metric-graph crossing probabilities were shown to converge to those formulas in the scaling limit (Karrila et al., 20 Jun 2026).

5. Higher-dimensional and transient-graph regimes

On transient weighted graphs $\tilde\phi$4 with polynomial volume growth exponent $\tilde\phi$5 and Green-function decay exponent $\tilde\phi$6, the cable-system GFF induces a bond-percolation model on the underlying discrete graph through the excursion set at level $\tilde\phi$7. In the regime

$\tilde\phi$8

the critical level is $\tilde\phi$9, and the critical one-arm probability satisfies

$\widetilde{\mathcal G}$0

More precisely, the upper bound is $\widetilde{\mathcal G}$1, where $\widetilde{\mathcal G}$2 is a slowly varying correction; for $\widetilde{\mathcal G}$3, this gives $\widetilde{\mathcal G}$4 up to a $\widetilde{\mathcal G}$5 factor, and for $\widetilde{\mathcal G}$6, $\widetilde{\mathcal G}$7 up to logarithmic corrections (Drewitz et al., 2023). The same work establishes sharp upper bounds for the truncated two-point function near criticality and identifies the correlation-length exponent $\widetilde{\mathcal G}$8 in this regime (Drewitz et al., 2023).

For the specific cable system $\widetilde{\mathcal G}$9, critical connectivity enjoys a quasi-multiplicativity property. When $C(x,y)$00,

$C(x,y)$01

while for $C(x,y)$02 there is a correction factor $C(x,y)$03, and $C(x,y)$04 is critical with bounds differing by $C(x,y)$05 (Cai et al., 2024). This result is a key input for the construction of the incipient infinite cluster.

For all $C(x,y)$06, $C(x,y)$07, four natural incipient infinite cluster constructions for the critical cable-system GFF coincide: conditioning on $C(x,y)$08 and letting $C(x,y)$09; conditioning on $C(x,y)$10 at supercriticality and letting the level tend to criticality; conditioning on $C(x,y)$11 and sending $C(x,y)$12; and conditioning on large cluster capacity (Cai et al., 2024). Under the resulting IIC law, the cluster is almost surely infinite and one-ended, and conditioned on $C(x,y)$13, the volume of the critical cluster in $C(x,y)$14 is typically of order

$C(x,y)$15

(Cai et al., 2024).

A further high-dimensional phenomenon is the microscopic separation of macroscopic sign clusters. For $C(x,y)$16, $C(x,y)$17, with uniformly positive probability there exist two distinct sign clusters of diameter at least $C(x,y)$18 in a box of size $C(x,y)$19 whose mutual distance is less than

$C(x,y)$20

in sharp contrast with the planar $C(x,y)$21 picture (Cai et al., 23 Oct 2025). As a byproduct, the number of pivotal edges for the one-arm event is typically of order

$C(x,y)$22

which implies that the cut-edge set of the IIC has dimension $C(x,y)$23 (Cai et al., 23 Oct 2025).

6. Loop soups, topological twists, and broader frameworks

The cable-system GFF is tightly linked to Brownian loop soups. On a metric graph, there is a coupling at loop-soup intensity $C(x,y)$24 such that the total loop local time equals $C(x,y)$25, and the loop-soup clusters coincide with the sign clusters of the metric-graph GFF (Aru et al., 2018, Cai et al., 2024). For first passage sets, Aru–Lupu–Sepúlveda showed that at level $C(x,y)$26 and nonnegative boundary data, the metric-graph FPS is exactly the union of loop-soup plus excursion clusters that contain a boundary excursion, together with the boundary itself (Aru et al., 2018). This yields a direct geometric interpretation of GFF level sets in terms of Brownian loops and excursions.

A further extension introduces gauge-twisted metric-graph GFFs. Given a $C(x,y)$27-valued gauge field $C(x,y)$28 on the edges, one defines a twisted Laplacian and a twisted GFF $C(x,y)$29. The usual metric-graph GFF conditioned on the topological event that every sign cluster is trivial for $C(x,y)$30 has the same absolute-value law as $C(x,y)$31, and the probability of this topological event is

$C(x,y)$32

where $C(x,y)$33 and $C(x,y)$34 are the untwisted and twisted Green matrices (Lupu, 2022). On annular planar domains this gives the probability that no sign cluster surrounds the hole (Lupu, 2022).

Finally, recent graph-theoretic work provides a dimension-free constructive viewpoint that is compatible with cable and metric settings. A discrete Hadamard variational formula expresses the Green function on a growing weighted graph as a sum of layer contributions, and the associated Hadamard operator gives a dynamic construction of the discrete GFF from white noise; the formulation is explicitly dimension-free and is stated to require no smoothness of any kind (Hedenmalm et al., 16 Mar 2026). A plausible implication is that similar Dirichlet-form-based constructions should extend to metric graphs and fractal geometries. Related approximation results for DGFFs on random graphs converging to compact manifolds show how graph Laplacians, Green operators, and Sobolev-space interpretations can be organized so that discrete GFFs converge to continuum GFFs (Cipriani et al., 2018).

Across these developments, the metric-graph GFF serves as a technically precise intermediary between lattice GFFs and continuum Gaussian free fields. Its continuity along edges is not a cosmetic refinement: it alters level-set connectivity, makes first passage sets pathwise meaningful, sharpens the loop-soup correspondence, and exposes both planar conformal structures and high-dimensional critical geometry in forms that are inaccessible in the purely vertex-based model.