Open problems from Werner's conformally invariant measure on self-avoiding loops: the polygon scaling limit, Brownian-loop structure, and Riemann surfaces

Werner (2008) constructed the unique (up to a multiplicative constant) measure $\mu$ on self-avoiding loops in the plane that satisfies conformal restriction, extended the construction to every Riemann surface as a family $(\mu_S)$, and stated a precise list of questions left open by the construction. One item has since been resolved: the explicit closed form of the annulus mass function $F(\rho)$ was proved by Ang, Remy and Sun (2022, published 2025), confirming a formula conjectured by Cardy (2006), up to the inherently non-canonical normalization constant. The surviving open core comprises: (i) the Lawler-Schramm-Werner conjecture (Conjecture 1 of the paper) that the lattice self-avoiding-polygon measures weighting $n$-step loops by $\lambda^{-n}$ converge as the mesh shrinks to a limiting measure on self-avoiding loops, a conformally invariant limit being necessarily a multiple of $\mu$ by Werner's uniqueness theorem; (ii) the connectivity of the component-neighbor graph of a planar Brownian loop and the multiplicative constants of the outer-boundary measures restricted to graph-distance events; (iii) the existence of a Brownian-loop-boundary description of $\mu_S$ on higher-genus surfaces; and (iv) a complete SLE-type loop-measure theory on arbitrary Riemann surfaces, for which the sphere theory of Kemppainen-Werner, Zhan's partial surface construction, and recent Liouville-quantum-gravity and uniqueness results give substantial but incomplete progress.

Background

A self-avoiding loop in the plane is the trace of a continuous injective map from the unit circle into the complex plane, viewed modulo reparametrization. Wendelin Werner constructed and uniquely characterized a canonical measure $\mu$ on this space. The characterizing property is conformal restriction: for any two conformally equivalent planar domains $D$ and $D'$, the image of $\mu$ restricted to the loops that stay in $D$, under any conformal map from $D$ onto $D'$, is exactly $\mu$ restricted to the loops that stay in $D'$. This forces $\mu$ to be invariant under translations, rotations and dilations, so $\mu$ has infinite total mass; Werner's Theorem 1 states that, up to multiplication by a positive constant, there is exactly one non-trivial measure on planar self-avoiding loops with this property. The same scheme defines a compatible family $(\mu_S)$ on every Riemann surface $S$ (a connected one-dimensional complex manifold; the plane and the Riemann sphere are the genus-zero cases, and a torus is the simplest higher-genus case), satisfying conformal invariance and restriction across surfaces.

The construction is probabilistic. The Brownian loop measure of Lawler and Werner is an infinite, conformally invariant measure on unrooted planar Brownian loops; the outer boundary of such a loop, meaning the boundary of the connected component of the complement that contains infinity, is almost surely a self-avoiding loop, and the outer-boundary measure induced by the Brownian loop measure is precisely $\mu$. Werner also studies the measure $N^{0}$ on Brownian loops rooted at the origin, normalized so that the $N^{0}$-mass of the set of loops with maximal radius between $1$ and $e$ equals $1$; the outer boundaries of loops under $N^{0}$ define a measure on self-avoiding loops surrounding the origin, and this measure is a constant multiple of $\nu$, the unique (up to a multiplicative constant) non-trivial measure supported on the self-avoiding loops surrounding the origin that satisfies conformal restriction in simply connected domains containing the origin. In the normalization where a Poissonian cloud of Brownian loops (a Brownian loop soup) of intensity $c$ corresponds to a statistical-physics model with central charge $c$, the same normalization fixes the constants between the Brownian objects and $\mu$. The measure $\mu$ is also tied to the Schramm-Loewner Evolution with parameter $\kappa=8/3$ (SLE$_{8/3}$), the random simple curve conjectured to describe scaling limits of self-avoiding-walk interfaces: loops sampled from $\mu$ are SLE$_{8/3}$-type curves, and $\mu$ arises as the degenerate $\kappa\to 8/3$ limit of the loop intensity measure of the conformal loop ensembles (CLE), the canonical random nested collections of non-intersecting loops that describe scaling limits of critical $O(n)$-model interfaces.

Two further objects frame the open questions. First, if $D$ is an annular region (a bounded doubly connected domain), then the $\mu$-mass of the set of loops that stay in $D$ and disconnect the two boundary components of $D$ depends only on the conformal modulus $\rho(D)$ of the annulus (in Werner's convention, $D$ is conformally equivalent to the round annulus $\{1\le|z|<e^{\rho}\}$); this mass is denoted $F(\rho)$. Events of this shape, namely that a loop stays in a given annular region and disconnects its two boundary components, generate the $\sigma$-field on which all the loop measures considered here are understood; it is the $\sigma$-field Werner works with throughout. Second, on the discrete side, a self-avoiding polygon on a periodic planar lattice $L$ is a closed lattice path visiting no vertex twice; the number of $n$-step self-avoiding walks grows like $\lambda^{n+o(1)}$ for a lattice-dependent connective constant $\lambda>1$, and assigning each $n$-step polygon the weight $\lambda^{-n}$ defines an infinite measure $\mu^{\delta}$ on loops drawn on the mesh-$\delta$ lattice $\delta L$. The structure of the connected components of the complement of a planar Brownian path has been studied since Le Gall's work on their cardinality and asymptotic shape, and Burdzy's theorem on cut points of planar Brownian paths (points of the path at which the past and the future of the motion lie in different connected components of the complement) implies that each complementary component of a Brownian loop has infinitely many neighboring components.

Werner's paper closes with a precise list of problems left open by these constructions. One of them, the explicit closed form of $F$, has since been resolved by Ang, Remy and Sun, who proved a formula conjectured by Cardy; the others remain open and constitute the problem stated below.

Problem Statement

The open core of Werner's question, after the external resolution of the annulus-formula item, is the following problem. The common scientific objective is to complete the identification and structural theory of the canonical conformally invariant self-avoiding-loop measure: its discrete lattice origin (part 1), its construction from Brownian loops in the plane (part 2), and its Brownian-loop and SLE-loop representation on Riemann surfaces (parts 3 and 4).

1. Discrete-to-continuum convergence (Conjecture 1 of the paper, attributed to Lawler-Schramm-Werner). For a regular periodic planar lattice $L$ with connective constant $\lambda$, let $\mu^{\delta}$ be the measure on self-avoiding loops drawn on the mesh-$\delta$ lattice $\delta L$ that assigns each $n$-step loop the weight $\lambda^{-n}$. Prove that $\mu^{\delta}$ converges as $\delta\to 0$ to a limiting measure on self-avoiding loops (in the sense of the annulus-event $\sigma$-field used by Werner, or in a stronger explicitly specified topology). By Werner's uniqueness theorem, if that limiting measure is conformally invariant, it is necessarily a multiple $c\,\mu$ of his measure.

2. Structure of the Brownian-loop complement. For a planar Brownian loop $Z$, define the component-neighbor graph of $Z$ as the graph whose vertices are the connected components of $\mathbb{C}\setminus Z$, with an edge between two components whenever some point of $Z$ lies on the boundary of both. Cut-point results imply that every component has infinitely many neighbors. Decide whether this graph is connected, i.e. whether any two complementary components can be joined by a finite chain of pairwise adjacent components. Furthermore, if $C_{0}$ and $C_{\infty}$ denote the components containing the origin and infinity and $A_{n}$ ($n\ge 1$) is the event that their distance in this graph is exactly $n$, then the law of the outer boundary of $Z$ under the Brownian loop measure restricted to $A_{n}$ is, by conformal restriction, a multiple of the measure $\nu$ on self-avoiding loops surrounding the origin ($\nu$ being, up to a multiplicative constant, the unique non-trivial measure on such loops satisfying conformal restriction in simply connected domains containing the origin, equivalently up to a constant the outer-boundary measure of Brownian loops rooted at the origin); compute these multiplicative constants. (For calibration: the corresponding constants for the restrictions to a fixed winding index follow from Yor's law for the Brownian loop index, and the unrestricted outer-boundary measure of loops surrounding the origin equals $\pi/5$ times the measure derived from $N^{0}$, Werner's measure on Brownian loops rooted at the origin (normalized so that the mass of loops with maximal radius between $1$ and $e$ is $1$), via the expected-area computation of Garban and Trujillo Ferreras; the $A_{n}$ constants are the open ones.)

3. Brownian-loop description on higher-genus surfaces. On a compact Riemann surface of genus at least one, the boundary of a complementary component of a Brownian loop need not be a simple curve, since double points occur with positive mass when the loop winds around the surface. Determine whether the measure $\mu_{S}$ on self-avoiding loops of such a surface nevertheless admits a direct measure-level description via boundaries of Brownian loops on $S$, for instance after restricting to loops all of whose complementary boundaries are simple, or after a suitable conditioning, and prove or disprove such a relation.

4. SLE-type loop measures on arbitrary Riemann surfaces. Construct an intrinsic theory of SLE-type loop measures on arbitrary Riemann surfaces, with existence, conformal invariance and the restriction property proved for all genera.

These four parts are stated as separate unresolved questions in the source: each is an independent open question, and a complete answer to any one part resolves that part without requiring the others. Within part 2, the connectivity decision and the computation of the constants are separate questions. The parts are linked by the single object $\mu$ and its Brownian-loop construction.

References

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