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arxiv: 2406.06522 · v2 · pith:5KO6XBV2new · submitted 2024-06-10 · 🧮 math-ph · math.MP· math.PR

Multiple SLEs for kappain (0,8): Coulomb gas integrals and pure partition functions

Yu Feng , Mingchang Liu , Eveliina Peltola , Hao Wu This is my paper

Pith reviewed 2026-05-23 23:52 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.PR
keywords SLECoulomb gas integralspartition functionsmeander matrixconformal field theorymultiple SLEloop O(n) models
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The pith

SLE partition functions are constructed as Coulomb gas integrals and related to pure partition functions by the meander matrix

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a family of SLE(κ) partition functions explicitly as Coulomb gas integrals for κ in (0,8). These integrals are positive for κ in (8/3,8) under their loop O(n) model interpretation and possess a Frobenius series expansion whose algebraic content matches that of conformal field theory. The authors further construct pure partition functions that are real-analytic for all κ in (0,8) and decay to zero like a polynomial in (8-κ) as κ approaches 8. They relate the two families of functions through the meander matrix. As a direct consequence the construction supplies global non-simple multiple chordal SLE(κ) measures for κ in (4,8) that are uniquely fixed by a re-sampling property.

Core claim

The Coulomb gas integrals serve as the SLE(κ) partition functions; when multiplied by the inverse of the meander matrix they produce the pure partition functions, which remain real-analytic on (0,8) and vanish polynomially as κ tends to 8. This explicit link also yields global non-simple multiple chordal SLE(κ) measures for κ in (4,8) that are determined solely by their re-sampling property.

What carries the argument

Coulomb gas integrals as SLE(κ) partition functions, related to pure partition functions via the meander matrix

If this is right

  • The partition functions remain positive throughout κ in (8/3,8)
  • They admit a Frobenius series expansion whose coefficients match the algebraic content of CFT
  • They display logarithmic asymptotics at the points κ=8/3 and κ=8
  • The meander-matrix relation produces global non-simple multiple chordal SLE(κ) measures for κ in (4,8) fixed by re-sampling

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit integral formulas may allow direct numerical evaluation of crossing probabilities for multiple SLE configurations
  • The polynomial vanishing rate near κ=8 quantifies how the measures degenerate when the curves become simple
  • Real-analytic dependence on κ permits continuation of SLE properties across the transition at κ=4

Load-bearing premise

The Coulomb gas integrals correspond to probabilistic correlations in loop O(n) models, which is invoked to prove positivity for κ in (8/3,8).

What would settle it

A direct numerical check that finds a negative value among the constructed Coulomb gas integrals for some κ strictly inside (8/3,8) would disprove the positivity statement.

Figures

Figures reproduced from arXiv: 2406.06522 by Eveliina Peltola, Hao Wu, Mingchang Liu, Yu Feng.

Figure 1.1
Figure 1.1. Figure 1.1: The O(n) model has an interesting phase diagram. This figure plots the critical edge-weight n 7→ pc(n) = (p 2 + √ 2 − n) −1 (in blue) and the critical edge-weight n 7→ pˆc(n) = (p 2 − √ 2 − n) −1 (in orange). Although the scaling limit when p > pc(n) is conjecturally the same, at the line ˆpc(n) one should see higher order corrections to the critical behavior at the lattice level [Nie82]. From the scalin… view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: In (a), we illustrate a boundary condition [PITH_FULL_IMAGE:figures/full_fig_p006_1_2.png] view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: The left panel depicts a plot of the function [PITH_FULL_IMAGE:figures/full_fig_p008_1_3.png] view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: The two pairs of Pochhammer integration contours [PITH_FULL_IMAGE:figures/full_fig_p009_1_4.png] view at source ↗
Figure 1.5
Figure 1.5. Figure 1.5: Plot of the fugacity κ 7→ ν(κ) (orange) and the constant κ 7→ C(κ) (blue). Note that ν(κ) = 0 at points of the form κ = 8 m with m ∈ 2Z≥0 + 1 odd, and ν(κ) ∈ {±2} at points of the form κ = 8 m with m ∈ 2Z>0 even. The constant C(κ) diverges at points of the form κ = 8 m with m ∈ 2Z>0 even (where ν(κ) ∈ {±2}). We prove Theorem 1.5 in Section 5 by combining analysis of the Coulomb gas integrals and multiple… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Illustration of the braid transformation [PITH_FULL_IMAGE:figures/full_fig_p027_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Illustration of Corollary 2.9: when κ = 8 2m−1 for m ∈ Z>0, the integral of f(x;u) along the Pochhammer contour ϑ(xar , xbr ) is the same as 2 times its integral along the simple clockwise loop ϱ(xar , xbr ). Corollary 2.9. Fix κ ∈  8 2m−1 : m ∈ Z>0 [PITH_FULL_IMAGE:figures/full_fig_p029_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Illustration of Lemma 2.10: when κ = 8 2m−1 for m ∈ Z>0, the integral of f(x;u) along the simple clockwise loop ϱr is the same as its integral along a loop ˚ϱr possibly surrounding more marked points. where Is ⊂ {1, 2, . . . , ℓ} \ {s}. Then, writing u = (u1, . . . , uℓ), we have ȷ ϱ1 du1 ȷ ϱ2 du2 · · · ȷ ϱℓ duℓ f(x;u) = ȷ ϱ1 du1 · · · ȷ ϱs−1 dus−1 ȷ ˚ϱs dus ȷ ϱs+1 dus+1 · · · ȷ ϱℓ duℓ f(x;u), (2.19) whe… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Example with N = 4 and ♭ = 2 in Definition 3.2. Proposition 3.3. Fix ♭ ∈ {1, 2, . . . , N}. The marginal law of ⃗η1 under P x1→x2♭ N can be described as follows. Let η be the chordal SLEκ on (H; x1, x2♭ ), and define the event A♭ (η) :=  η ∩ (x2s, x2s+1) = ∅ for all 1 ≤ s ≤ ♭ − 1 [PITH_FULL_IMAGE:figures/full_fig_p039_3_1.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Illustration of the property (4.9) in Lemma 4.8. [PITH_FULL_IMAGE:figures/full_fig_p051_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Illustration of the relation (4.14). 4.3 General M¨obius covariance We next strengthen the M¨obius covariance (Lemma 4.9) into a covariance property under any M¨obius map φ ∈ PSL(2, R) preserving the upper half-plane, which may rotate the marked boundary points. Lemma 4.9. Fix κ ∈ (4, 8) and α ∈ LPN . The function Z⃗ α of Definition 4.1 satisfies the M¨obius covariance Z⃗ α(x) = Y 2N i=1 |φ ′ (xi)| h × Z… view at source ↗
read the original abstract

In this article, we give an explicit relationship of SLE partition functions with Coulomb gas formalism of conformal field theory. We first construct a family of SLE$(\kappa)$ partition functions as Coulomb gas integrals and derive their various properties. In accordance with an interpretation as probabilistic correlations in loop $O(n)$ models, they are always positive when $\kappa \in (8/3,8)$, while they may have zeroes for $\kappa \le 8/3$. They also admit a Frobenius series expansion that matches with the algebraic content from CFT. Moreover, we check that at the first level of fusion, they have logarithmic asymptotic behavior when $\kappa = 8/3$ and $\kappa = 8$, in accordance with logarithmic minimal models $M(2,1)$ and $M(2,3)$, respectively. Second, we construct $\SLE_\kappa$ pure partition functions and show that they are real-analytic in $\kappa \in (0,8)$ and decay to zero as a polynomial of $(8-\kappa)$ as $\kappa \to 8$. We explicitly relate the Coulomb gas integrals and pure partition functions together in terms of the meander matrix. As a by-product, our results yield a construction of global non-simple multiple chordal SLE$(\kappa)$ measures ($\kappa \in (4,8)$) uniquely determined by their re-sampling property.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript constructs a family of SLE(κ) partition functions as Coulomb gas integrals for κ∈(0,8), establishes their positivity for κ∈(8/3,8) via an interpretation as correlations in loop O(n) models, derives Frobenius series expansions matching CFT algebraic content, and verifies logarithmic asymptotics at the first level of fusion for κ=8/3 and κ=8. It further constructs SLE_κ pure partition functions that are real-analytic in κ∈(0,8) and decay polynomially in (8−κ) as κ→8, relates the two families explicitly via the meander matrix, and obtains as a by-product global non-simple multiple chordal SLE(κ) measures for κ∈(4,8) uniquely determined by the re-sampling property.

Significance. If the constructions and relations hold, the work supplies explicit integral representations that connect multiple SLE partition functions to the Coulomb gas formalism of CFT, together with analytic continuation properties in κ and links to logarithmic minimal models. The explicit meander-matrix relation between the Coulomb gas integrals and the pure partition functions, together with the resulting construction of global SLE measures, would constitute a concrete advance in the rigorous theory of multiple SLEs.

major comments (2)
  1. [Abstract] Abstract (first paragraph): the claim that the constructed Coulomb gas integrals 'are always positive when κ∈(8/3,8)' is justified exclusively by their interpretation as probabilistic correlations in loop O(n) models. Because this positivity is invoked both to assert the sign and to underwrite the probabilistic meaning required for the global non-simple SLE measures, the manuscript should supply a self-contained argument establishing positivity directly from the integral representation (e.g., sign of the integrand after suitable regularization or explicit evaluation at special points) rather than relying on the external loop-model dictionary.
  2. [Section on pure partition functions and meander-matrix relation] The section constructing the pure partition functions and their relation to the Coulomb gas integrals (the second main part of the paper): the explicit relation via the meander matrix is load-bearing for the uniqueness claim on the global SLE measures; the manuscript should state the precise form of this linear relation (including the range of κ for which the matrix is invertible) and verify that the resulting objects satisfy the re-sampling property without additional assumptions.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'at the first level of fusion' for the logarithmic asymptotics is used without a brief definition or reference to the corresponding fusion rule in the Coulomb gas integrals; adding one sentence would improve clarity for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (first paragraph): the claim that the constructed Coulomb gas integrals 'are always positive when κ∈(8/3,8)' is justified exclusively by their interpretation as probabilistic correlations in loop O(n) models. Because this positivity is invoked both to assert the sign and to underwrite the probabilistic meaning required for the global non-simple SLE measures, the manuscript should supply a self-contained argument establishing positivity directly from the integral representation (e.g., sign of the integrand after suitable regularization or explicit evaluation at special points) rather than relying on the external loop-model dictionary.

    Authors: The positivity statement in the abstract is explicitly tied to the standard correspondence between the Coulomb gas integrals and correlations in the loop O(n) model, which is a well-established dictionary in the SLE/CFT literature. Our integrals are constructed precisely to reproduce these quantities, so the positivity is inherited from the probabilistic model rather than derived ab initio from the integral representation. A fully self-contained sign analysis of the multi-dimensional integrals (independent of the loop-model interpretation) would require substantial additional analytic work that lies outside the scope of the present paper. We will revise the abstract and the relevant introductory paragraph to make this reliance explicit and to note that the probabilistic interpretation supplies the positivity. revision: partial

  2. Referee: [Section on pure partition functions and meander-matrix relation] The section constructing the pure partition functions and their relation to the Coulomb gas integrals (the second main part of the paper): the explicit relation via the meander matrix is load-bearing for the uniqueness claim on the global SLE measures; the manuscript should state the precise form of this linear relation (including the range of κ for which the matrix is invertible) and verify that the resulting objects satisfy the re-sampling property without additional assumptions.

    Authors: The manuscript already presents the explicit linear relation between the Coulomb gas integrals and the pure partition functions via the meander matrix in the section on pure partition functions. We will expand this presentation to include the precise matrix entries (or their defining combinatorial formula) and to state the range of invertibility, which holds for all κ ∈ (0,8) except at the discrete degeneracy points where the central charge takes rational values corresponding to minimal models. The re-sampling property for the resulting global measures follows directly from the characterizing properties of the pure partition functions (analyticity, polynomial decay, and the fusion rules encoded by the meander matrix) together with the standard construction of multiple SLE measures from such partition functions; no extra assumptions are required. We will add a short verification paragraph making this dependence explicit. revision: yes

Circularity Check

0 steps flagged

No circularity; constructions use standard external formalisms

full rationale

The paper constructs SLE partition functions explicitly as Coulomb gas integrals and relates them to pure partition functions via the meander matrix from prior literature. Positivity for κ∈(8/3,8) is asserted by reference to an external loop O(n) model interpretation rather than derived internally from the integrals themselves. No quoted equations reduce by construction to fitted inputs or self-citations, and the central claims (analyticity, decay, global measures) remain independent of any self-referential chain. This is the normal case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based solely on abstract; no explicit free parameters, axioms, or invented entities are identifiable beyond standard background tools of Coulomb gas integrals and meander matrix from prior work.

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Forward citations

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · cited by 1 Pith paper · 2 internal anchors

  1. [1]

    Pole dynamics and an integral of motion for multiple SLE(0)

    [ABKM20] Tom Alberts, Sung-Soo Byun, Nam-Gyu Kang, and Nikolai Makarov. Pole dynamics and an integral of motion for multiple SLE(0). Preprint in arXiv:2011.05714,

    work page arXiv 2011

  2. [2]

    Conformal welding of quantum disks and multiple SLE: the non-simple case

    [AHSY23] Morris Ang, Nina Holden, Xin Sun, and Pu Yu. Conformal welding of quantum disks and multiple SLE: the non-simple case. Preprint in arXiv:2310.20583,

    work page arXiv

  3. [3]

    Conformal random geometry

    [Dup06] Bertrand Duplantier. Conformal random geometry. In Mathematical Statistical Physics: Lecture Notes of the Les Houches Summer School (2005) . Elsevier,

    work page 2005

  4. [4]

    Preprint in arXiv:2310.10234,

    and an application for critical FK-Ising model. Preprint in arXiv:2310.10234,

    work page arXiv

  5. [5]

    Standard modules, radicals, and the valenced Temperley-Lieb algebra

    [FP18] Steven M. Flores and Eveliina Peltola. Standard modules, radicals, and the valenced Temperley-Lieb algebra. Preprint in arXiv:1801.10003,

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Flores and Eveliina Peltola

    [FP20] Steven M. Flores and Eveliina Peltola. Higher spin quantum and classical Schur-Weyl duality for sl(2). Preprint in arXiv:2008.06038,

    work page arXiv 2008

  7. [7]

    Flores, Jacob J

    [FSK15] Steven M. Flores, Jacob J. H. Simmons, and Peter Kleban. Multiple-SLE κ connectivity weights for rectangles, hexagons, and octagons. Preprint in arXiv:1505.07756,

    work page arXiv

  8. [8]

    Boundary correlations in planar LERW and UST

    [KKP20] Alex Karrila, Kalle Kyt¨ ol¨ a, and Eveliina Peltola. Boundary correlations in planar LERW and UST. Comm. Math. Phys. , 376(3):2065–2145,

    work page 2065

  9. [9]

    101 [LSW01a] Gregory F

    Preprint in arXiv:2108.04421. 101 [LSW01a] Gregory F. Lawler, Oded Schramm, and Wendelin Werner. The dimension of the planar Brownian frontier is 4 /3. Math. Res. Lett., 8:401–411,

    work page arXiv

  10. [10]

    ICMP, Montreal, July 2018)

    Special issue (Proc. ICMP, Montreal, July 2018). [Pel20] Eveliina Peltola. Basis for solutions of the Benoit & Saint-Aubin PDEs with particular asymptotic properties. Ann. Inst. Henri Poincar´ e D, 7(1):1–73,

    work page 2018

  11. [11]

    Critical percolation in the plane

    (Updated 2009, see arXiv:0909.4499). [Smi06] Stanislav Smirnov. Towards conformal invariance of 2 D lattice models. In Proceedings of the ICM 2006, Madrid, Spain , volume 2, pages 1421–1451. European Mathematical Society,

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  12. [12]

    Conformal loop ensembles: The Markovian character- ization and the loop-soup construction

    [SW12] Scott Sheffield and Wendelin Werner. Conformal loop ensembles: The Markovian character- ization and the loop-soup construction. Ann. Math., 176(3):1827–1917,

    work page 1917

  13. [13]

    SLE partition functions via conformal welding of random surfaces

    [SY23] Xin Sun and Pu Yu. SLE partition functions via conformal welding of random surfaces. Preprint in arXiv:2309.05177,

    work page arXiv

  14. [14]

    Random planar curves and Schramm-Loewner evolutions

    [Wer03] Wendelin Werner. Random planar curves and Schramm-Loewner evolutions. Lecture notes from the Saint-Flour Summer School (July 2002),

    work page 2002

  15. [15]

    Commutation relations for two-sided radial SLE

    103 [WW24] Yilin Wang and Hao Wu. Commutation relations for two-sided radial SLE. Preprint in arXiv:2405.07082,

    work page arXiv

  16. [16]

    Existence and uniqueness of nonsimple multiple SLE

    [Zha23] Dapeng Zhan. Existence and uniqueness of nonsimple multiple SLE. Preprint in arXiv: 2308.13886,

    work page arXiv