Axiomatic characterization of non-simple conformal loop ensembles and a loop-soup construction for $\kappa \in (4,8)$

Sheffield and Werner (2012) proved that random collections of simple, disjoint, non-nested loops in a planar simply connected domain satisfying non-triviality, conformal invariance, and the conformal restriction property are exactly the conformal loop ensembles $\mathrm{CLE}_\kappa$ with $\kappa \in (8/3,4]$, and they constructed these ensembles as the outer boundaries of outermost clusters of a subcritical Brownian loop soup. For $\kappa \in (4,8)$, the ensembles $\mathrm{CLE}_\kappa$ built from Sheffield's branching $\mathrm{SLE}_\kappa(\kappa-6)$ exploration trees consist of non-simple, self-touching loops that may touch one another and the domain boundary. It is now known that these ensembles are well defined and do satisfy the natural nondisjoint variant of the axioms, but the two linked questions posed in the final section of Sheffield–Werner (2012) remain open: (1) are the $\mathrm{CLE}_\kappa$, $\kappa \in (4,8)$, the only loop collections satisfying that variant, which would extend the Markovian characterization to the regime conjecturally describing scaling limits of critical FK (Fortuin–Kasteleyn) random-cluster interfaces, including percolation ($\mathrm{CLE}_6$); and (2) is there an analog of the loop-soup construction in the non-simple regime, given that the Brownian loop soup produces only the simple ensembles.

Background

Schramm–Loewner evolution ($\mathrm{SLE}_\kappa$) is the one-parameter family of random fractal curves in a planar simply connected domain that are, loosely speaking, the only random curves with a conformally invariant law and independent increments when parametrized by capacity; the parameter $\kappa > 0$ controls roughness ($\kappa \le 4$ gives simple curves, $\kappa > 4$ gives self-touching curves). A conformal loop ensemble ($\mathrm{CLE}_\kappa$) is the loop-level counterpart: a random countable collection of non-crossing loops, nested inside one another, with a conformally invariant law. For each $\kappa \in (8/3,8)$ there is, up to scale normalization, a canonical such ensemble, constructed by Sheffield (2009) by gluing together a branching tree of $\mathrm{SLE}_\kappa(\kappa-6)$ processes — that is, $\mathrm{SLE}_\kappa$ with an additional force point at the starting point. This is called the exploration-tree construction. For $\kappa \in (8/3,4]$ every loop is a simple curve, distinct loops are disjoint, and they stay at positive distance from the domain boundary; for $\kappa \in (4,8)$ the loops are non-simple: each is quasisimple (a uniform limit of simple loops, so it may touch but not cross itself), distinct loops may touch one another, and loops may touch the domain boundary.

Sheffield and Werner (2012) characterized the simple regime axiomatically. A non-nested simple loop configuration is a countable collection of pairwise disjoint, non-nested simple loops such that only finitely many loops have diameter above any fixed positive threshold (local finiteness). Their axioms on such a random collection $\Gamma$ in a simply connected domain $D$ are: non-triviality ($\Gamma$ is nonempty with positive probability); conformal invariance (the law of $\Gamma$ is unchanged under conformal maps of $D$); and the conformal restriction property (for every simply connected subdomain $U \subseteq D$, conditionally on the loops of $\Gamma$ that are not contained in $U$, the loops lying in the domain $U^*$ obtained from $U$ by filling in the holes cut off by those loops are distributed as independent copies of $\Gamma$, conformally mapped into the connected components of $U^*$). Their main theorem states that these axioms hold exactly for the outermost loops of $\mathrm{CLE}_\kappa$ with $\kappa \in (8/3,4]$, and that each loop is then an $\mathrm{SLE}_\kappa$-type loop. They also proved a loop-soup construction: the Brownian loop soup of Lawler and Werner — a Poissonian, conformally invariant ensemble of outer boundaries of planar Brownian loops with intensity $c \in (0,1]$ — has clusters whose outermost outer boundaries form $\mathrm{CLE}_\kappa$ with $c = (3\kappa-8)(6-\kappa)/(2\kappa)$, so the loop soup produces precisely the simple ensembles; for $c > 1$ there is almost surely a single cluster, so no non-simple ensemble arises this way.

For $\kappa \in (4,8)$ the situation is different. Sheffield (2009) constructed $\mathrm{CLE}_\kappa$ via the exploration tree but did not prove its starting-point independence there, reducing it to a reversibility conjecture for $\mathrm{SLE}_\kappa$ (Conjecture 3.11 in that paper); his Theorem 5.4 characterizes the outermost loops of $\mathrm{CLE}_\kappa$, $\kappa \in (4,8)$, under five axioms that include an explicit $\mathrm{SLE}(\kappa)$ conformal Markov property — a much stronger hypothesis than the Sheffield–Werner restriction axioms. Miller and Sheffield's imaginary-geometry program — a framework that couples $\mathrm{SLE}_\kappa$ to the Gaussian free field, a random generalized function on the plane — later proved reversibility of $\mathrm{SLE}_\kappa$ for $\kappa \in (4,8)$, making the construction well defined, and Gwynne–Miller–Qian proved conformal invariance of whole-plane $\mathrm{CLE}_\kappa$ on the Riemann sphere in this regime; the conformal restriction property and local finiteness are likewise documented (Miller–Schoug). Miller–Sheffield–Werner's continuum-percolation theory establishes a duality coupling each non-simple $\mathrm{CLE}_{\kappa'}$, $\kappa' \in (4,6)$, with a simple $\mathrm{CLE}_{16/\kappa'}$, and shows that non-simple loops have no cut points, so their outer boundaries are simple loops. What is missing are the two statements asked in the final section of Sheffield–Werner (2012): a characterization by the weak, restriction-type axioms, and a loop-soup-type construction. The non-simple regime is the one relevant to scaling limits of critical FK (Fortuin–Kasteleyn) random-cluster interfaces — a dependent-percolation model on the lattice, parameterized by a cluster weight $q$, which reduces to Bernoulli percolation at $q = 1$ and to the Ising model at $q = 2$: $\mathrm{CLE}_{\kappa'}$, $\kappa' \in (4,8)$, is the conjectured (and in key cases proved) scaling limit of the loop ensemble of the FK$_q$ model with $q = 4\cos^2(4\pi/\kappa')$, which includes ordinary critical percolation ($\kappa' = 6$, $q = 1$) and the FK-Ising model ($\kappa' = 16/3$, $q = 2$).

Problem Statement

Let $D \subsetneq \mathbb{C}$ be a simply connected domain, and let $\Gamma$ be a random countable collection of loops in $D$ (loops viewed as equivalence classes of continuous maps from $[0,1]$ to $\overline{D}$ modulo monotone reparametrization) satisfying the following hypotheses, which form the natural nondisjoint variant of the Sheffield–Werner axioms:

• Loop configuration: every loop of $\Gamma$ is quasisimple (a uniform limit of simple loops, so it may touch but not cross itself); no loop of $\Gamma$ lies in the bounded component of the complement of another loop (the collection is non-nested, i.e., these are the outermost loops); and for each $\varepsilon > 0$ only finitely many loops have diameter exceeding $\varepsilon$ (local finiteness). Unlike in the simple regime, loops may touch themselves, and $\Gamma$ is required to be genuinely non-disjoint: almost surely, loops of $\Gamma$ intersect one another and the boundary $\partial D$.
• Non-triviality: $\Gamma$ is nonempty with positive probability.
• Conformal invariance: for every conformal map $\phi$ from $D$ onto another simply connected domain, $\phi(\Gamma)$ has the same law as $\Gamma$.
• Conformal restriction: for every simply connected $U \subseteq D$ with $U \ne D$, conditionally on the loops of $\Gamma$ that are not contained in $U$, the remaining loops are distributed as independent copies of $\Gamma$, conformally mapped into the connected components of the domain obtained from $U$ by filling in the holes cut off by those loops.

The open problem of Sheffield–Werner (2012, Section 12), both parts of which remain open, is:

(1) Characterization (uniqueness). Prove that $\Gamma$ must then be the collection of outermost loops of $\mathrm{CLE}_\kappa$ for some $\kappa \in (4,8)$ — thereby extending the Sheffield–Werner Markovian characterization theorem from $\kappa \in (8/3,4]$ to the nondisjoint regime — or exhibit a $\Gamma$ satisfying the above axioms that is not the outermost-loop ensemble of any $\mathrm{CLE}_\kappa$. The converse direction, that the $\mathrm{CLE}_\kappa$ with $\kappa \in (4,8)$ satisfy these axioms, is already known, so the open content is uniqueness. A characterization whose axioms include an SLE-type conformal Markov property (as in Sheffield's five-axiom Theorem 5.4) does not extend this Markovian characterization, since it builds the SLE structure into the hypotheses; moreover, the precise formulation of the nondisjoint restriction axiom is itself part of the open problem.

(2) Loop-soup construction. Construct, for $\kappa \in (4,8)$, an analog of the Brownian loop-soup construction: a conformally invariant Poissonian random ensemble of loops from which $\mathrm{CLE}_\kappa$ is obtained as the outermost boundaries of clusters, or via another explicitly described operation, with the parameter correspondence identified — or prove that no such construction exists. The Brownian loop soup produces only the simple ensembles (intensity $c \in (0,1]$ corresponding to $\kappa \in (8/3,4]$) and a single cluster for $c > 1$, so a genuinely new mechanism is required.

Both parts jointly constitute the source question: extending the Markovian characterization and the loop-soup construction of conformal loop ensembles to the nondisjoint case.

References

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