Brownian Loop-Soups on Cable-Graphs

• The paper establishes that Brownian loop-soups on cable-graphs form Poisson ensembles where a critical intensity of 1/2 governs percolation and scaling limits.
• The methodology employs metric graph construction to analyze Brownian motion, precise arm exponents, and coupling with Gaussian free fields across dimensions.
• The study reveals singularity in conditional loop laws and an intensity doubling phenomenon in high dimensions, enhancing our understanding of universality in random media.

Brownian loop-soups on cable-graphs are Poissonian ensembles of Brownian loops defined on metric graphs, which play a central role in the scaling theory of random processes, statistical mechanics, and the geometry of Gaussian free fields (GFFs) on discrete structures. The cable-graph, also called a metric graph, is formed by replacing each combinatorial edge of a discrete graph (such as $\mathbb{Z}^d$) with a continuous line segment (“cable”) of corresponding length, supporting continuous-time Brownian motion that interpolates between discrete and differential structures. Properties of Brownian loop-soups (BLS) on cable-graphs encode fine aspects of percolation, isomorphism identities, scaling limits, singularity phenomena, and critical behavior, with deep connections to arm exponents, occupation measures, and GFF cluster geometry.

1. Construction of Brownian Loop-Soups on Cable-Graphs

The cable-graph associated to a discrete graph $G = (V,E)$ is constructed by replacing each edge $e = \{x,y\} \in E$ by a line segment $I_e$ of prescribed length, typically $1$, connecting $x$ and $y$. The resulting metric space $\Gamma$ admits a natural one-dimensional Brownian motion: inside each cable $I_e$, the process evolves as standard Brownian motion; at a vertex $v$, it chooses uniformly among incident cables to continue. Given an underlying transient discrete graph, the Brownian motion on $\Gamma$ is also transient, and reversible with respect to the sum of Lebesgue measure on cables and counting measure on vertices.

The (unrooted) Brownian loop measure $\mu$ on $\Gamma$ integrates over all starting points, durations, and Brownian bridges, giving

$$\mu(d\gamma) = \int_{z\in\Gamma} \int_0^\infty P^t_{z,z}(d\gamma) p_t(z,z) \frac{dt}{t} dz,$$

where $P^t_{z,z}$ is the law of the Brownian bridge of duration $t$ from $z$ back to $z$, and $p_t(z,z)$ its transition density.

A Brownian loop-soup $\mathcal{L}^\lambda$ at intensity $\lambda > 0$ is a Poisson point process on the space of continuous loops in $\Gamma$ with intensity measure $\lambda\mu$. The critical intensity, at which macroscopic loop clusters or percolation transitions appear, is $\lambda_c = 1/2$ (Lupu et al., 26 Nov 2025, Werner, 10 Feb 2025). In the planar case ($d = 2$), one focuses on the intensity $\alpha=1/2$, which is also critical for the associated GFF coupling.

2. Cluster Geometry, Criticality, and Dimensional Exponents

Clusters in the Brownian loop-soup are defined as maximally connected unions of loops whose traces intersect. As the intensity parameter increases, clusters grow and percolation phenomena arise. At the critical intensity $\lambda=1/2$, the geometry of clusters is tightly connected to fundamental critical exponents dictating long-distance connectivity, cluster volume, and fractal dimension.

For the two-dimensional cable-graph over $\mathbb{Z}^2$, pivotal quantities are the probabilities $\pi_4(k,n)$ and $\pi_2^+(k,n)$ for "four-arm" and "two-arm" crossing events: the existence of two well-separated BLS clusters crossing an annulus $A_{k,n}$, or a boundary cluster crossing a half-annulus $A^+_{k,n}$, respectively. At $\alpha=1/2$, these probabilities obey sharp power laws: $$\pi_4(k,n) \asymp (k/n)^2, \qquad \pi_2^+(k,n) \asymp (k/n),$$ establishing bulk arm exponent $2$ and boundary arm exponent $1$ (Bi et al., 29 Sep 2025).

In higher dimensions, particularly on the cable-graph over $\mathbb{Z}^d$, $d \geq 3$, the cluster volume and box-counting dimension at criticality are governed by the scaling of one-arm and crossing probabilities. For $d=3$, the clusters of the metric-graph BLS have almost surely upper box-counting dimension $5/2$, whereas in the continuum Brownian loop soup on $\mathbb{R}^3$, the cluster dimension is strictly less than $5/2$ (Cai et al., 23 Oct 2025). This dimension gap arises from the contribution and connectivity structure of microscopic edge-loops in the metric context, which persist in the scaling limit.

3. Coupling to Gaussian Free Field and Switching Identities

A central theme is the Le Jan isomorphism: at critical intensity, the occupation time field $$\Lambda(x) = \sum_{\beta \in \mathcal{L}} \ell_\beta(x)$$, where $\ell_\beta(x)$ is the local time of loop $\beta$ at $x$, is distributed as $1/2\,\varphi(x)^2$, where $\varphi$ is the cable-graph GFF with appropriate boundary (Bi et al., 29 Sep 2025, Werner, 10 Feb 2025). The clusters of BLS coincide exactly with the sign clusters of the GFF. Lupu’s coupling makes this correspondence exact, allowing cluster events in the loop model to be analyzed via GFF techniques.

Switching identities reveal that, conditioning two points $x,y$ to be in the same cluster (or sign cluster), the conditional law of the occupation-time field is that of the unconditioned loop-soup plus an odd number of independent Brownian excursions between $x$ and $y$ (Werner, 10 Feb 2025). This partitioning permits explicit computation of large-scale correlations and critical exponents, and underpins the understanding of percolation and scaling limits.

4. Singularity Phenomena in Conditional Laws

An exceptional property of BLS on cable-graphs is the singularity of the conditioned law of individual loops, given the total occupation field $\Lambda$. Specifically, for a c-loop-soup on $\Gamma$ with occupation field $\Lambda$, the conditional law of any individual loop $\beta_n$, conditioned on $\Lambda$, is singular with respect to the original Brownian loop measure $\mu$ (Dremaux, 4 Dec 2025). This arises because "fast points" of $\Lambda$—points where $\Lambda$ exhibits rapid local variation—force every contributing loop through such points to display non-typical local time increments, i.e., the occupation density cannot satisfy the standard Lévy modulus. This non-disintegrability has significant implications for the probabilistic structure of the loop-soup and its Markovian couplings to the GFF.

Key consequences include the impossibility of absolute continuous disintegration of the occupation field into individual loops, and the necessity of employing rewiring or switching identities rather than conditional densities in Markovian constructions.

5. Intensity Doubling and Universality in High Dimensions

In dimensions $d \geq 7$, the critical BLS on the cable-graph exhibits an "intensity doubling" phenomenon for macroscopic cycles: proper self-avoiding cycles of diameter comparable to $N$ occur with limiting law equivalent to the superposition of two independent BLS of critical intensity (Lupu et al., 26 Nov 2025). Proper cycles can be generated either as the trace of a single large loop, or as a chain of small loops ("ghost soup"), with asymptotically equal probability $1/2$. In GFF language, the macroscopic cycles in the sign clusters converge, in the scaling limit, to a Brownian loop-soup of twice the critical intensity.

This is formalized via the switching property for cluster indices: the parity (even/odd) of winding numbers around a fixed $(d-2)$-plane in high dimensions is perfectly balanced, guaranteeing that each macroscopic cluster can realize either scenario—containing a genuine macroscopic loop or being formed purely by small loops—with probability approaching $1/2$.

A summary of intensity doubling is as follows:

Regime ($d$)	Macroscopic cycle origin	Limiting law
$d\geq7$	Large single loop ("big-loop")	BLS at intensity $1/2$
$d\geq7$	Chain of small loops ("ghost soup")	Independent BLS at intensity $1/2$ (“doubled” total)

This doubling is a principal indicator of mean-field universality in high dimensions, matching predictions from Bernoulli percolation.

6. Arm Events, Critical Exponents, and Related Percolation Models

The analysis of multi-arm events—such as four-arm and two-arm events in planar or half-plane domains—is fundamental for understanding the geometry of BLS clusters. The precise power-law asymptotics for these arm probabilities in the 2D cable-graph,

$$\pi_4(k,n) \asymp (k/n)^2,\qquad \pi_2^+(k,n) \asymp(k/n),$$

were verified using comparison to GFF connection events and reflection principle estimates for the GFF (Bi et al., 29 Sep 2025). The separation lemma and quasi-multiplicativity are used to relate crossing events at different scales.

These exponent bounds also transfer to the random walk loop soup (RWLS) on $\mathbb{Z}^2$ via stochastic domination, establishing the universality class across different discretizations. A key application is to the percolation of low-level sets of the discrete GFF, where the existence of crossings with the field bounded by $a\sqrt{\log\log N}$ (for large enough $a$) relies on these arm events.

7. Scaling Limits, Dimensional Gaps, and Outlook

A key insight is that metric-graph BLS and their clusters do not always converge, under scaling, to their continuum Brownian loop-soup analogs—for example, in $d=3$ the cluster dimensions reveal a strict gap, rooted in the persistent influence of small cable loops that are absent in the continuum model (Cai et al., 23 Oct 2025). In dimensions $4,5$, the conjecture is that appropriate "gluing" mechanisms, deterministic or random, are needed to recover cable-graph cluster characteristics from the continuum limit.

Open questions include the precise description of such gluing in low and intermediate dimensions and the characterization of phase transitions for percolation in continuum BLS in physically relevant dimensions. The singularity results (Dremaux, 4 Dec 2025) and universality features in high dimensions (Lupu et al., 26 Nov 2025) point to a rich landscape of regimes distinguished by the interplay between microscopic loop structure, Markovian couplings, and global cluster geometry.