SLE₈/₃ Loop Measure: Conformal Invariance in Random Geometry

• SLE₈/₃ loop measure is the conformally invariant infinite measure on simple, self-avoiding loops, underpinning the scaling limits of self-avoiding walks.
• It is constructed via Brownian loop soups, branching SLE, and conformal welding, demonstrating key properties like restriction, conformal invariance, and Markovian structure.
• The measure's explicit formulas, moment computations, and connections to CLE, Liouville quantum gravity, and spectral geometry offer deep insights into random planar systems.

The SLE$_{8/3}$ loop measure is the conformally invariant infinite measure on simple (self-avoiding) loops in the plane or on Riemann surfaces, canonically associated to the Schramm–Loewner Evolution (SLE) with parameter $\kappa = 8/3$. It is central to the rigorous description of scaling limits of self-avoiding walks (SAW) and related statistical physics models and plays a foundational role in the theory of conformal loop ensembles (CLE), Brownian loop soups, Liouville quantum gravity (LQG), and logarithmic conformal field theory (LCFT).

1. Definition, Characterizations, and Construction

The SLE$_{8/3}$ loop measure can be defined through several equivalent frameworks:

• Brownian Loop Soup Construction: The outer boundaries of clusters in a Poissonian Brownian loop soup of intensity $c$ in a simply connected planar domain $D$ yield countable collections of simple disjoint loops. When $c = 1$, the outer boundaries become SLE$_\kappa$ loops for $\kappa = 4$; for $0 < c < 1$, the correspondence $\kappa \in (8/3,4]$, $c = \frac{(3\kappa-8)(6-\kappa)}{2\kappa}$ associates to each $c$ a unique $\kappa$ (Sheffield et al., 2010). At the lower end, as $c \downarrow 0$, one isolates the law of a single simple loop, giving the SLE$_{8/3}$ loop measure.
• Branching SLE Construction: The loop ensemble traced by collections of branching SLE$_\kappa$ curves, with $\kappa = 8/3$, also yields the SLE$_{8/3}$ loop measure in the scaling limit of SAW (Sheffield et al., 2010).
• Werner’s Measure: Werner’s canonical infinite measure on self-avoiding loops, satisfying strict conformal restriction (Higgs et al., 7 Jan 2024), is normalized so that for nested domains $U \subset V \subset \mathbb{C}$ and a marked point $0 \in U$, the measure of loops in $V \setminus \{0\}$ surrounding $0$ but not contained in $U$ is $\log |\varphi'(0)|$, where $\varphi: U \to V$ is conformal with $\varphi(0)=0$.
• CLE Intensity and Conformal Welding: The loop intensity measure of full-plane CLE$_\kappa$ for $\kappa = 8/3$ (interpreted in the limit) coincides, up to normalization, with the SLE$_{8/3}$ loop measure and can be realized canonically as the interface created by conformal welding of two independent Liouville quantum gravity disks with $\gamma = \sqrt{8/3}$ (Ang et al., 2022, Ang et al., 25 Sep 2024).
• Minkowski Content Approach: Rooted and unrooted SLE$_\kappa$ loop measures on the Riemann sphere $\widehat{\mathbb{C}}$ are constructed by unweighting two-sided whole-plane SLE$_\kappa$ curves by their natural Minkowski content (with Hausdorff dimension $4/3$ for $\kappa = 8/3$) (Zhan, 2017).

All approaches yield an infinite, $\operatorname{PGL}(2, \mathbb{C})$-invariant measure on noncrossing simple loops, uniquely determined (up to constant) by conformal restriction and Möbius invariance.

2. Restriction, Conformal Invariance, and Markov Properties

The restriction property is fundamental: for any two simply connected domains $U \subset V$ with $0 \in U \subset V$, the measure of loops in $V \setminus \{0\}$ that surround $0$ but are not contained in $U$ is $\log |\varphi'(0)|$. This normalization completely determines the measure up to overall constant (Higgs et al., 7 Jan 2024).

SLE$_{8/3}$ loop measure enjoys the following:

• Conformal Invariance: The measure is preserved under biholomorphic maps, making it natural for scaling limits of conformally invariant statistical mechanics models (Sheffield et al., 2010, Zhan, 2017, Higgs et al., 7 Jan 2024).
• Restriction: Conditioning on loops contained in subdomains restricts the measure conformally (Sheffield et al., 2010).
• Markovian Structure: The measure is compatible with explorations of the loops (as in branching SLE descriptions) (Sheffield et al., 2010).
• Space-Time Homogeneity: When parametrized by Minkowski content, the (rooted) loop measure is invariant under time shifts and re-rootings (Zhan, 2017).

These properties guarantee the universality of SLE$_{8/3}$ loop measure as the scaling limit of self-avoiding planar loop ensembles.

3. Explicit Formulas and Probabilistic Observables

Explicit computations for probabilities and moments associated with SLE$_{8/3}$ loops provide rigorous physical predictions:

• Two-Point Functions and Crossing Probabilities: The probability that a chordal SLE$_{8/3}$ passes to the left of points $z, w$ is given by

$$\mathbb{P}\{\gamma \text{ passes left of } z,w\} = \left[\tfrac{1}{2} + \tfrac{x}{2|z|}\right] \left[\tfrac{1}{2} + \tfrac{u}{2|w|}\right]\left[1 + \frac{y}{x + |z|} \frac{v}{u + |w|} G(\sigma)\right]$$

where $\sigma = \frac{|z-w|^2}{|z-\overline{w}|^2}$ and $$G(\sigma) = 1 - \sigma \cdot { }_2F_1(1,4/3;5/3;1-\sigma)$$ (Beliaev et al., 2010).

• Bubble and Area Statistics: For SLE$_{8/3}$ bubbles conditioned to be macroscopic, explicit area and area-moment formulas are derived, and their numerical evaluation closely supports conjectures that area is Airy-distributed (Beliaev et al., 2010). For instance, $\mathbb{E}[\text{area}]=\pi/10$ and $\mathbb{E}[(\text{area})^2] \approx \pi/30$.
• Cardy’s Formula for Annuli: The total measure of homotopically nontrivial loops in a standard annulus, as predicted by Cardy and supported by rigorous computations, is given by

$$F_0(\rho) = 6\pi \left[\sum_{k \in \mathbb{Z}} (-1)^{k-1} k q^{3k^2/2 - k + 1/8} \right] \prod_{n=1}^\infty (1-q^n)$$

with $q = \exp(-2\pi^2 / \rho)$ (Higgs et al., 7 Jan 2024).

• Moment Generating Functions: The moment generating function for the electrical thickness $\theta(\eta)$ of a loop sampled from the shape measure is given exactly, for $\lambda < 1 - \kappa/8$,

$$\mathbb{E}[e^{\lambda \theta(\eta)}] = \frac{\sin(\pi(1-\kappa/4))}{\pi(1-\kappa/4)} \cdot \frac{\pi \sqrt{(1-\kappa/4)^2 + \lambda \kappa/2}}{\sin\left( \pi \sqrt{(1-\kappa/4)^2 + \lambda\kappa/2} \right)}$$

(Ang et al., 25 Sep 2024).

4. Loop Measure, CLE, and Loop Soups

The SLE$_{8/3}$ loop measure underlies the law of the conformal loop ensemble CLE$_{8/3}$, which is the unique conformally invariant probability measure on countable collections of disjoint, simple, noncrossing planar loops (Sheffield et al., 2010, Sheffield et al., 2010, Zhan, 2017). CLE$_{8/3}$ can be viewed as:

• Scaling limit of self-avoiding loop ensembles: As the mesh is sent to zero, the scaling limit of critical $O(n=0)$ loop models or SAW interfaces converges to CLE$_{8/3}$, underpinned by the SLE$_{8/3}$ loop measure (Alekseev, 2021).
• Brownian Loop Soup Outer Boundaries: In a planar domain, at $c = 0$ (lower boundary), the Brownian loop soup yields a trivial configuration; for $0 < c < 1$ the outer boundaries of clusters are simple SLE$_{\kappa}$-type loops (Sheffield et al., 2010), and in the $c \to 0$ limit, one recovers the measure on single simple loops.
• CLE Intensity and Loop Selection: The intensity (i.e., the generator of Poissonian selection of loops) coincides, after normalization, with the SLE$_{8/3}$ loop measure (Ang et al., 25 Sep 2024).

CLE$_{8/3}$ and its nesting properties (i.e., the statistics of how many loops surround a point at small scales) are governed by fractal-multifractal laws, and their exceptional sets (e.g., points surrounded by atypically many loops) have computable Hausdorff dimensions (Miller et al., 2013).

5. Liouville Quantum Gravity, Conformal Welding, and Duality

Conformal welding of independent Liouville quantum gravity (LQG) disks, each parameterized by a quantum boundary length, gives a direct and canonical construction of the SLE$_{8/3}$ loop measure: the interface between the two quantum disks follows the law of an SLE$_{8/3}$ loop (Ang et al., 2022, Ang et al., 25 Sep 2024). This construction is fundamental for:

• Link to Random Maps: The interface of random planar maps decorated with a self-avoiding polygon converges, after scaling, to SLE$_{8/3}$ loop on the LQG sphere (Brownian map) (Ang et al., 2022).
• Exact Formulae and Constants: Using LQG and Liouville conformal field theory (LCFT), precise constants (reflection coefficients, disk correlation functions, DOZZ formulae) are computed for SLE loop measures (Ang et al., 25 Sep 2024).
• Duality: There is a duality at the level of loop measures between SLE loop measures for $\kappa$ and $16/\kappa$. For instance, the outer boundary of a sample from the SLE$_{16/\kappa}$ loop measure has the law of the SLE$_{\kappa}$ loop measure, up to a multiplicative constant (Ang et al., 25 Sep 2024).

6. Analytical, Spectral, and Geometric Aspects

• Brownian Loop Measure Decomposition: The Brownian loop measure on Riemann surfaces admits an explicit decomposition by free homotopy class, connecting loop statistics to the length spectrum of closed geodesics: $$\mu_X(\mathcal{C}_X(\gamma)) = \frac{1}{m(\gamma)} \cdot \frac{1}{e^{\ell_\gamma} - 1}$$ where $m(\gamma)$ is the multiplicity and $\ell_\gamma$ the hyperbolic length (Wang et al., 13 Jun 2024).
• Spectral Invariants and Zeta-Regularized Determinants: Renormalization of the total mass of the Brownian loop measure yields formulas for the determinant of the Laplacian in terms of geometric data (lengths of geodesics), highlighting deep connections between random loops and spectral geometry (Wang et al., 13 Jun 2024).
• Loewner Energy and Onsager–Machlup Functional: The Loewner energy $I^{\mathrm{L}}(\Gamma)$ of a Jordan curve $\Gamma$ is the Onsager–Machlup large-deviation rate function for the SLE$_{8/3}$ loop measure (Wang, 2018, Fan, 9 Aug 2025). The relative measure of small neighborhoods of loops is governed by their energy difference, yielding a variational principle: $$\lim_{\varepsilon \to 0} \frac{Q^\kappa(O_\varepsilon(\gamma))}{Q^\kappa(O_\varepsilon(\gamma_0))} \approx \exp\left\{ \frac{c(\kappa)}{24}[I^{\mathrm{L}}(\gamma) - I^{\mathrm{L}}(\gamma_0)] \right\}$$ where $c(\kappa)$ is the central charge.

7. Further Properties, Applications, and Open Directions

• Explicit Evaluation and Numerics: Cardy’s conjecture for the measure of nontrivial loops in an annulus is numerically supported via Schwarz–Christoffel computations of conformal moduli (Higgs et al., 7 Jan 2024).
• Connection to LCFT: SLE$_{8/3}$-related observables (passage probabilities, Green’s functions) correspond to correlation functions in $c=0$ logarithmic CFT, facilitating explicit calculations via the Coulomb gas formalism and operator product expansions (Alekseev, 2021).
• Future Directions: The SLE$_{8/3}$ loop measure’s precise normalization and its connection to LCFT structure constants (including the imaginary DOZZ formula) continue to generate active research in random geometry, spectral theory, and quantum gravity (Ang et al., 25 Sep 2024).
• CLE Nesting and GFF Extremes: Nesting statistics of SLE$_{8/3}$ loops mirror extreme value statistics of the two-dimensional Gaussian Free Field (GFF), establishing quantitative parallels between CLE multifractality and GFF thick points (Miller et al., 2013, Miller et al., 2013).

The SLE$_{8/3}$ loop measure thus functions as a uniquely characterized and universal measure for random simple planar loops at criticality, informed by deep interconnections between probability, geometry, analysis, quantum gravity, and conformal field theory.