Computing the constant $\alpha(\kappa)$ in Kemppainen–Werner's theorem $\nu^i=\nu^e=\alpha(\kappa)\,\nu^{\mathrm{cle}}$ for the nested simple conformal loop ensembles on the Riemann sphere

Kemppainen and Werner (Probab. Theory Relat. Fields 165, 2016) constructed the nested full-plane conformal loop ensemble $\mathrm{CLE}_\kappa$ on the Riemann sphere for $\kappa\in(8/3,4]$ and proved its invariance under the inversion $z\mapsto 1/z$. Their Theorem 2 states that the intensity measures $\nu^i$ and $\nu^e$ of the inner and outer boundaries of clusters of a full-plane $\mathrm{SLE}_{8/3}$ loop soup that surround the origin are equal, and both equal $\alpha(\kappa)$ times the loop intensity measure $\nu^{\mathrm{cle}}$ of the full-plane nested CLE — but the finite positive constant $\alpha(\kappa)$ is left completely undetermined, and the Minkowski-content lemma underpinning the proof is established only along a subsequence. In the current Liouville-quantum-gravity-based integrability literature $\alpha(\kappa)$ remains uncomputed: recent work identifies $\nu^{\mathrm{cle}}$ with Zhan's SLE loop measure up to constants and computes several CLE-side constants while explicitly leaving the loop-measure normalizations open. The problem asks for a closed-form computation of $\alpha(\kappa)$ with the source's exact measure normalizations.

Background

Schramm–Loewner evolution ($\mathrm{SLE}_\kappa$) is the one-parameter family of random fractal curves arising as scaling limits of interfaces in two-dimensional critical lattice models; for $\kappa<4$ the curve is almost surely simple, and for $\kappa<8$ it has Hausdorff (and Minkowski) dimension $d=1+\kappa/8$. The conformal loop ensemble ($\mathrm{CLE}_\kappa$) is the corresponding canonical random collection of nested, non-crossing loops in a planar domain, defined for $\kappa\in(8/3,8)$; for $\kappa\in(8/3,4]$ the loops are simple, and the ensemble can be constructed from a Brownian loop soup — a Poisson point process of Brownian loops with intensity $c\in(0,1]$, related to $\kappa$ by the Sheffield–Werner loop-soup construction through $c=(3\kappa-8)(6-\kappa)/(2\kappa)$ — by taking the outer boundaries of the outermost loop-soup clusters. Replacing each Brownian loop by its outer boundary yields the $\mathrm{SLE}_{8/3}$ loop soup, a Poisson process of simple SLE-type loops that, unlike the Brownian loop soup itself, is invariant in law under all Möbius transformations (compositions of translations, rotations, dilations, and the inversion $z\mapsto 1/z$) of the Riemann sphere $\hat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$.

Kemppainen and Werner constructed the nested full-plane $\mathrm{CLE}_\kappa$ for $\kappa\in(8/3,4]$ as the local limit of nested CLEs in growing discs, and proved it is invariant in distribution under inversion, hence under all Möbius transformations (their Theorem 1). The proof runs through their Theorem 2, which compares two infinite measures on loops surrounding the origin: the intensity measures $\nu^i$ and $\nu^e$ of, respectively, the inner and outer boundaries of the clusters of a full-plane $\mathrm{SLE}_{8/3}$ loop soup that surround the origin — defined concretely by $\nu^i(A)=\mathbb{E}[\#\{j:\gamma^i_j\in A\}]$ and analogously for $\nu^e$, where $\gamma^i_j,\gamma^e_j$ are the inner and outer boundaries of the $j$-th surrounding cluster — and the intensity measure $\nu^{\mathrm{cle}}$ of the loops surrounding the origin in the full-plane nested CLE. Theorem 2 asserts

$\nu^i=\nu^e=\alpha(\kappa)\,\nu^{\mathrm{cle}}$

for some finite positive constant $\alpha(\kappa)$, whose value the paper does not determine or characterize. Heuristically, $\alpha(\kappa)$ is the number of inner (equivalently outer) loop-soup cluster boundaries surrounding a point per nested-CLE loop surrounding that point, per unit logarithmic scale; it is a well-defined absolute constant because both normalizations are fixed by counting (expectations of counting measures), not by convention.

The analytic heart of the proof compares the areas $|V^i_\varepsilon(\gamma)|$ and $|V^e_\varepsilon(\gamma)|$ of the inner and outer $\varepsilon$-neighborhoods of a cluster-boundary loop $\gamma$ (the sets of points in the $\varepsilon$-neighborhood of $\gamma$ that lie inside, respectively outside, $\gamma$, with $|\cdot|$ denoting Euclidean area, as in the source). Lawler and Rezaei proved that for chordal $\mathrm{SLE}_\kappa$ curves the $d$-dimensional Minkowski content — the limit as $\varepsilon\downarrow 0$ of $\varepsilon^{d-2}$ times the $\varepsilon$-neighborhood area — exists and defines the natural length parametrization of the curve (the parametrization in which unit time corresponds to unit Minkowski content, the fractal analogue of arclength); Kemppainen and Werner's Lemma 3 (in the published numbering; Lemma 11 in arXiv v2) establishes the analogous statement for the cluster-boundary loops only along a deterministic subsequence $\varepsilon_n\downarrow 0$: for $\nu^e$-almost every loop $\gamma$,

$\lim_{n\to\infty}\varepsilon_n^{d-2}|V^e_{\varepsilon_n}(\gamma)|=\lim_{n\to\infty}\varepsilon_n^{d-2}|V^i_{\varepsilon_n}(\gamma)|=L(\gamma)\in(0,\infty).$

They remark that the full limit (as $\varepsilon\downarrow 0$, rather than along a subsequence) could also be derived, but is not needed for their purposes, and they do not write the argument.

The constant $\alpha(\kappa)$ sits at the active interface between CLE, Liouville quantum gravity (LQG, the random two-dimensional geometry built from the Gaussian free field, the canonical Gaussian random surface on a planar domain), and the recent integrability program for CLE. Zhan constructed $\mathrm{SLE}_\kappa$ loop measures on the Riemann sphere for all $\kappa\in(0,8)$ via the Minkowski content of SLE and an un-rooting procedure; Ang, Cai, Sun and Wu proved that for $\kappa\in(8/3,8)$ the CLE loop intensity measure (the $\nu^{\mathrm{cle}}$ of Kemppainen–Werner, which they explicitly attribute to the 2016 paper) agrees with Zhan's measure up to a multiplicative constant, and computed several multiplicative constants attached to the CLE intensity measure, while stating that they do not know how to compute the corresponding constants for Zhan's loop measure. No existing result determines the constant linking the $\mathrm{SLE}_{8/3}$ loop-soup side (the measures $\nu^i$, $\nu^e$) to the CLE side — which is exactly $\alpha(\kappa)$.

Problem Statement

Within the framework and notation of Kemppainen–Werner (2016), for each $\kappa\in(8/3,4]$ (equivalently, each full-plane $\mathrm{SLE}_{8/3}$ loop-soup intensity $c=c(\kappa)\in(0,1]$), complete Theorem 2 quantitatively. The objective is the exact identification of the loop-soup cluster-boundary intensity measures with the CLE loop intensity measure on the Riemann sphere: compute $\alpha(\kappa)$ in closed form. That is, give an explicit expression for the constant in

$\nu^i=\nu^e=\alpha(\kappa)\,\nu^{\mathrm{cle}}$

— as a function of $\kappa$ (or of the soup intensity $c$), possibly involving named special-function values or constants from the Liouville quantum gravity / CLE integrability literature — and prove, with all three measures normalized exactly as in the source ($\nu^i$, $\nu^e$ as expectations of counting measures over the inner/outer boundaries of full-plane $\mathrm{SLE}_{8/3}$ loop-soup clusters surrounding the origin, and $\nu^{\mathrm{cle}}$ as the loop intensity measure of the full-plane nested CLE), that the constant of Theorem 2 equals this value.

The sought result is the exact value of the constant under the source's normalizations: not merely the existence of some proportionality constant, and not the identification of the measures only up to an unspecified multiplicative constant. The non-subsequential two-sided Minkowski-content limit described in the Background is a gap the source itself flags as derivable but leaves unwritten; it is recorded here as context rather than as a requested result. Two adjacent directions from the same source — alternative derivations of inversion invariance (e.g. via the full-plane Gaussian free field) and full-plane discrete-to-continuum convergence of critical O(n)/Ising models — are likewise outside the scope of this problem: they are open research programs without a single completion boundary rather than parts of the same quantitative objective.

References

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