Coulomb-gas partition functions for multiple SLEs at exceptional parameters: probabilistic, logarithmic-CFT, and lattice interpretation

The source paper (Feng, Liu, Peltola, Wu, arXiv:2406.06522) constructs multiple- $\mathrm{SLE}_\kappa$ partition functions as Coulomb-gas screening integrals for all $\kappa\in(0,8)$ , relates them to pure partition functions through the meander matrix, and proves that their Frobenius (fusion) asymptotics acquire logarithmic terms at $\kappa=8/3$ and $\kappa=8$ , matching logarithmic-CFT predictions. It explicitly leaves open what these objects mean at the exceptional parameters $\kappa=8/m$ : there the normalization is not defined, the raw integrals vanish on the subfamily $\kappa=8,8/3,8/5,\ldots$ , and the meander matrix degenerates. The problem is to complete this exceptional-parameter picture: give the renormalized partition functions and renormalized meander-matrix relations a probabilistic or CFT interpretation (in particular at $\kappa=8/3,8/5,8/7,\ldots$ ); identify the indecomposable Virasoro/Temperley-Lieb module structure underlying the logarithmic operator-product behavior at the logarithmic minimal points $\kappa=8/3$ ( $c=0$ ) and $\kappa=8$ ( $c=-2$ ); determine higher-order Frobenius terms and exact coefficients at general rational $\kappa=4p/p'$ ; and derive the (renormalized) meander-matrix connection-probability relations from discrete loop $O(n)$ models in general polygons beyond the proven integrable cases, notably the self-avoiding-walk asymptotics at $\kappa=8/3$ .

Background

Schramm-Loewner evolution ( $\mathrm{SLE}_\kappa$ ) is the one-parameter family of random conformally invariant curves that describe, rigorously in special cases and conjecturally in general, the scaling limits of critical interfaces in two-dimensional lattice models. For $2N$ marked boundary points $x_1<\cdots<x_{2N}$ of a simply connected domain, multi-curve analogs are encoded by multiple- $\mathrm{SLE}_\kappa$ partition functions: smooth positive functions on the configuration space satisfying the Belavin-Polyakov-Zamolodchikov (BPZ) equations - second-order null-state partial differential equations which in conformal field theory (CFT) express the degeneracy of a primary field - together with Möbius covariance. A distinguished basis of this solution space is given by the pure partition functions $\mathcal{Z}^{(\kappa)}_\alpha$ of Kytölä and Peltola, indexed by planar link patterns $\alpha\in\mathrm{LP}_N$ , i.e. non-crossing pairings of the $2N$ boundary points; they govern the marginal laws of the individual curves and the connection probabilities between link patterns.

The Coulomb-gas formalism of Dotsenko and Fateev produces CFT correlation functions by inserting screening charges (vertex operators in a background charge) and integrating their positions along contours. The source paper defines, for each link pattern $\beta$ , a Coulomb-gas integral $\mathcal{F}^{(\kappa)}_\beta$ : a normalized contour integral of $N$ screening charges of charge $-2/\sqrt{\kappa}$ around arcs joining the pairs of $\beta$ , with an integrand built from powers of $(u_j-x_i)$ ; formally this is a correlation function of $2N$ copies of the level-two degenerate primary field $\Phi_{1,2}$ of conformal weight $h(\kappa)=(6-\kappa)/(2\kappa)$ in a free-boson background. The paper proves that the $\mathcal{F}^{(\kappa)}_\beta$ are multiple- $\mathrm{SLE}_\kappa$ partition functions for all $\kappa\in(0,8)$ , positive exactly on the intervals $\kappa\in\bigcup_m(8/(4m-1),\,8/(4m-3))$ - and in particular on the range $\kappa\in(8/3,8)$ relevant to critical loop $O(n)$ models - and that they are related to the pure partition functions by the meander matrix (studied in this context by Di Francesco, Golinelli, and Guitter): $\mathcal{F}^{(\kappa)}_\beta=\sum_\alpha M(\alpha,\beta)\,\mathcal{Z}^{(\kappa)}_\alpha$ , where $M(\alpha,\beta)=\nu^{\ell(\alpha,\beta)}$ counts the loops $\ell(\alpha,\beta)$ of the meander obtained by superimposing the link patterns $\alpha$ and the reflection of $\beta$ , weighted by the Temperley-Lieb loop weight $\nu(\kappa)=-2\cos(4\pi/\kappa)$ (the Temperley-Lieb algebra being the diagram algebra of non-crossing pairings on the marked boundary points, the algebraic backbone of the loop $O(n)$ model) - the same parameter that weights loops in the loop $O(n)$ model with $n=\nu$ .

Everything degenerates at the exceptional values $\kappa=8/m$ , $m=1,2,3,\ldots$ (equivalently, $\kappa\in\{4/m\}\cup\{8/(2m+1)\}\cup\{8\}$ ), where the paper's normalization constant is not well-defined: on the vanishing family $\kappa\in\{8\}\cup\{8/(2m+1)\}$ (odd $m$ ) the Coulomb-gas integrals vanish identically, the loop weight $\nu(\kappa)$ equals zero, and the meander matrix degenerates. The paper rescues the objects by renormalization: at $\kappa\in\{8/(2m+1)\}$ the rescaled limits $\widehat{\mathcal{F}}^{(\kappa)}_\beta$ (Proposition 1.7), and at $\kappa=8$ the limits $\widehat{\mathcal{F}}^{(8)}_\beta$ (Proposition 1.8), are positive multiple- $\mathrm{SLE}_\kappa$ partition functions expressible through the renormalized meander matrix $\widehat{M}(\alpha,\beta)=\lim_{\nu\to0}\nu^{-1}M(\alpha,\beta)=1\{\text{the meander of }\alpha,\beta\text{ has one loop}\}$ (its Eq. (1.29) and Proposition 1.15; at $\kappa=8$ the relation reads $\widehat{\mathcal{F}}^{(8)}_\beta=\sum_\alpha\widehat{M}(\alpha,\beta)\,\widehat{\mathcal{Z}}^{(8)}_\alpha$ with the renormalized pure partition functions). But the paper leaves their meaning open, stating that it would be interesting to find probabilistic or CFT interpretations for these partition functions.

The physics expectation is logarithmic CFT. Solutions of the BPZ equations admit Frobenius-type asymptotic expansions as two insertion points collide (fusion), and at the borderline values $\kappa=8/3$ (central charge $c=0$ ; dilute $O(0)$ , i.e. self-avoiding walk; logarithmic minimal model $M(2,3)$ ) and $\kappa=8$ ( $c=-2$ ; dense $O(0)$ , i.e. the uniform spanning tree; $M(2,1)$ ) these expansions contain logarithmic terms (Propositions 1.18 and 1.19), exactly reproducing the logarithmic operator-product expansions predicted by Gurarie for the stress tensor and its logarithmic partner. More generally the exceptional values $\kappa=8/m$ correspond to central charges $c(8/m)=13-6(2/m+m/2)=1-6(p-p')^2/(pp')$ with $m=2p'/p$ , labeling the logarithmic minimal models $M(p,p')$ of Pearce, Rasmussen, and Zuber; the $O(n)$ -model CFTs are expected to be logarithmic for all $n\in(0,2)$ (Gorbenko and Zan; Read and Saleur). A logarithmic CFT is a non-diagonal theory whose Virasoro modules are indecomposable (fields form Jordan blocks under $L_0$ ) rather than irreducible. What is missing mathematically is the identification of which indecomposable modules the renormalized Coulomb-gas integrals realize correlation functions on, with exact structure constants; the source notes in particular that no mathematically complete treatment of the $\mathrm{SLE}_{8/3}$ /stress-energy-tensor relationship (observed by Friedrich and Werner) is available.

On the lattice side, the loop $O(n)$ model on the hexagonal lattice assigns weight $n$ per loop and $p$ per occupied edge; Nienhuis predicted its critical line and critical exponents; the parameterization $n=-2\cos(4\pi/\kappa)$ relates these critical points to the conjectural $\mathrm{SLE}_\kappa$ /CLE $_\kappa$ (conformal loop ensemble, the loop-valued analogue of SLE) scaling limits in the dilute ( $\kappa\in(8/3,4]$ ) and dense ( $\kappa\in[4,8)$ ) phases. The paper's Conjecture 1.2 (also predicted in the physics literature by Flores, Simmons, Kleban, and Ziff) asserts that the scaling-limit connection probabilities of the interfaces in a polygon with boundary condition $\beta$ are $\lim_{\delta\to0}\mathbb{P}[\mathrm{conn}^\delta=\alpha]=M_n(\alpha,\beta)\,\mathcal{Z}^{(\kappa)}_\alpha/\mathcal{F}^{(\kappa)}_\beta$ . This is proven only for the integrable cases: percolation ( $n=1$ , $p=1$ ; Cardy-Smirnov at $N=2$ ), the Ising model ( $n=1$ at $p_c$ ; Peltola-Wu), the FK-Ising (Fortuin-Kasteleyn random-cluster) model ( $n=\sqrt2$ ; Feng-Peltola-Wu), and the uniform spanning tree ( $n\to0$ ; Liu-Peltola-Wu, with the renormalized meander matrix). For the self-avoiding walk ( $n\to0$ dilute, $\kappa=8/3$ ) the paper conjectures the partition-function asymptotics $\delta^{-5N/4}\sum_\omega p_c(0)^{o(\omega)}\approx\mathcal{Z}^{(8/3)}_\alpha$ (its Equation (1.6)), and the only known SAW-to-SLE $_{8/3}$ convergence is in the Liouville quantum gravity setting (Gwynne-Miller). Rigorous results on the loop $O(n)$ model itself remain far from conformal invariance for general $n$ (see the survey of Peled and Spinka).

Status of the corners: the $\kappa=8$ (uniform spanning tree, $c=-2$ ) corner of the interpretation questions has been substantially resolved by the companion work of Liu-Peltola-Wu (Coulomb-gas $\mathrm{SLE}_8$ partition functions as UST Peano-curve partition functions, crossing probabilities as their ratios, and a match to the Virasoro staggered-module classification) and extended by Karrila-Lafay-Peltola-Roussillon to fused branches with a valenced Temperley-Lieb algebra action; the $\kappa=4$ ( $c=1$ ) corner is developed by Lafay-Peltola-Roussillon and Karrila-Peltola-Schoug through degenerate $c=1$ conformal blocks (fused Specht polynomials, a fusion-adapted polynomial basis built from symmetric-group Specht modules) and Gaussian-free-field level lines. The genuinely open core is the odd-denominator exceptional family $\kappa=8/(2m+1)$ , $m\ge1$ (starting at $\kappa=8/3$ ), the general rational $\kappa=4p/p'$ degeneracy structure, and the lattice derivation beyond the integrable cases.

Problem Statement

Overarching objective: complete the exceptional-parameter picture of the multiple- $\mathrm{SLE}_\kappa$ Coulomb-gas partition functions constructed in the source paper, by answering with mathematical rigor the four linked questions it leaves open. The common objects are the Coulomb-gas integrals $\mathcal{F}^{(\kappa)}_\beta$ , their renormalizations at the exceptional parameters $\kappa=8/m$ , the pure partition functions $\mathcal{Z}^{(\kappa)}_\alpha$ , and the (renormalized) meander matrix relating them; the common question is what these objects are - analytically, representation-theoretically, and probabilistically - at the degenerate parameters. The four facets are:

(i) Interpretation at exceptional parameters. For the exceptional values $\kappa=8/m$ at which the construction degenerates - in particular the vanishing family $\kappa\in\{8\}\cup\{8/(2m+1):m\ge1\}$ where the renormalized integrals and the renormalized meander matrix are defined - give the renormalized partition functions and the renormalized meander-matrix relations $\widehat{\mathcal{F}}^{(\kappa)}_\beta=\sum_\alpha\widehat{M}(\alpha,\beta)\,\mathcal{Z}^{(\kappa)}_\alpha$ (at $\kappa=8$ : $\widehat{\mathcal{F}}^{(8)}_\beta=\sum_\alpha\widehat{M}(\alpha,\beta)\,\widehat{\mathcal{Z}}^{(8)}_\alpha$ , with the renormalized pure partition functions) a concrete probabilistic meaning (for example, as partition functions or connection probabilities of an identified lattice or continuum model) or a CFT meaning (correlation functions in an identified conformal field theory), beyond the known $\kappa=8$ uniform-spanning-tree case - in particular for $\kappa=8/3,8/5,8/7,\ldots$

(ii) Representation-theoretic identification of the logarithmic structure. At the logarithmic minimal points $\kappa=8/3$ ( $c=0$ , dilute $O(0)$ , $M(2,3)$ ) and $\kappa=8$ ( $c=-2$ , dense $O(0)$ , $M(2,1)$ ), and generally at the $M(2,p')$ points $\kappa=8/(2m+1)$ , beyond the known $\kappa=8$ case, identify the indecomposable Virasoro (or Temperley-Lieb / extended-algebra) modules on which the renormalized Coulomb-gas integrals realize boundary correlation functions, so that the logarithmic Frobenius terms become operator-product logarithms with explicitly computed structure constants, in the spirit of Gurarie's predicted operator-product expansions; provide, in particular, the mathematically complete treatment of the $\mathrm{SLE}_{8/3}$ /stress-energy-tensor relationship whose absence the source notes.

(iii) Frobenius expansions at general rational parameters. Extend the Frobenius-series analysis - exact higher-order terms and coefficients, beyond the leading terms already established at $\kappa\in(8/3,8)\setminus\{8/m\}$ , $\kappa=8/3$ , $\kappa=8$ , and $\kappa=4$ - to the general rational values $\kappa=4p/p'$ (with $p,p'$ coprime positive integers) and to the intervals $\kappa\in(8/(4m-1),\,8/(4m-3))$ for $m\ge2$ , where the source expects the same results to hold but reports the detailed analysis to become increasingly complicated.

(iv) Lattice derivation. Prove the scaling-limit connection-probability formula for critical loop $O(n)$ interfaces in general polygons with $2N$ marked boundary points (the source's Conjecture 1.2, stated there for all $n\in(0,2]$ ) for values of $n$ beyond the proven integrable cases (percolation, Ising, FK-Ising, uniform spanning tree), including the renormalized meander-matrix relations at exceptional parameters; in particular, establish the self-avoiding-walk asymptotics conjectured in the source's Equation (1.6), namely the convergence of the $N$ -walk connectivity partition functions (normalized by $\delta^{-5N/4}$ and the critical edge weight) to the pure partition functions $\mathcal{Z}^{(8/3)}_\alpha$ .

The facets are scientifically linked: they concern the same function family at the same degenerate parameters and jointly inform what the exceptional- $\kappa$ multiple-SLE partition functions are. A complete resolution of any one of the four facets (i)-(iv), or of the $\kappa=8/3$ ( $c=0$ ) case of (i)-(ii), suffices as a complete resolution of this task, since each facet is an independently resolvable open question; the four facets together constitute the full exceptional-parameter program.

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