Toward a conformal field theory for Schramm-Loewner evolutions

Peltola, Eveliina

doi:10.1063/1.5094364

Cited by 13 publications

(9 citation statements)

References 149 publications

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“…Second, the proof of Theorem 2.1 also provides an alternative proof showing that the partition functions Z N and Z α satisfy the PDEs that appear in the definition of the so-called local multiple SLE partition functions (see, e.g., [KP16, Appendix A]). The same PDEs appear in Conformal field theory as degeneracy PDEs for correlation functions of primary fields [Pel19]. A different proof for the theorem below was given in [KKP17, Theorem 4.1] by a direct computation based on the explicit expressions (4.3) and (4.2).…”

Section: Sle Type Processesmentioning

confidence: 97%

“…, known or conjectured to describe the scaling limits of random interfaces in many critical planar lattice models [Smi01, LSW04, SS05, CN07, Zha08, SS09, HK13, CDCH + 14, Izy15,GW18]. A particularly interesting variant is the local multiple SLE [Dub07,KP16], which explicitly connects SLEs to Conformal field theory, the physics description of scaling limits of critical models [BBK05,Gra07,Dub15,KKP19,Pel19]. The main result of this article, Theorem 2.1, proves local multiple SLE convergence for multiple boundary-to-boundary branches in a uniform spanning tree (UST) model on Z 2 , as well as its natural generalization to other isoradial lattices.…”

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confidence: 99%

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UST branches, martingales, and multiple SLE(2)

Karrila¹

2020

Preprint

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We identify the local scaling limit of multiple boundary-to-boundary branches in a uniform spanning tree (UST) as a local multiple SLE(2), i.e., an SLE(2) process weighted by a suitable partition function. By recent results, this also characterizes the "global" scaling limit of the full collection of full curves. The identification is based on a martingale observable in the UST with N branches, obtained by weighting the well-known martingale in the UST with one branch by the discrete partition functions of the models. The obtained weighting transforms of the discrete martingales and the limiting SLE processes, respectively, only rely on a discrete domain Markov property and (essentially) the convergence of partition functions. We illustrate their generalizability by sketching an analogous convergence result for a boundary-visiting UST branch and boundary-visiting SLE(2).

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Section: Sle Type Processesmentioning

confidence: 97%

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confidence: 99%

UST branches, martingales, and multiple SLE(2)

Karrila¹

2020

Preprint

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“…We have used the parameterization by κ to make connection with SLE κ theory (see [Pel19] for references). For example, we have h 1,1 (κ) = 0, h 1,2 (κ) = 6−κ 2κ , and h 1,3 (κ) = 8−κ κ .…”

Section: Speculation: Logarithmic Cft For Ustmentioning

confidence: 99%

“…In light of conformal field theory, these properties are to be expected (cf. [Pel19]), and the main difficulty is to verify them rigorously. Of these properties, the PDEs (B.2) follow from a martingale argument for the interface, and many of the other properties are natural from the construction.…”

Section: B2 Proof Of Theorem B1 and Corollary B2mentioning

confidence: 99%

Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=-2$ Logarithmic CFT

Liu,

Peltola,

2021

Preprint

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We give a direct probabilistic construction for correlation functions in a logarithmic conformal field theory (log-CFT) of central charge −2. Specifically, we show that scaling limits of Peano curves in the uniform spanning tree in topological polygons with general boundary conditions are given by certain variants of the SLE κ with κ = 8. We also prove that the associated crossing probabilities have conformally invariant scaling limits, given by ratios of explicit SLE 8 partition functions. These partition functions are interpreted as correlation functions in a log-CFT. Remarkably, it is clear from our results that this theory is not a minimal model and exhibits logarithmic phenomena -indeed, the limit functions have logarithmic asymptotic behavior, that we calculate explicitly. General fusion rules for them could be inferred from the explicit formulas.

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“…A different probabilistic approach to conformal invariance has been developed during the past twenty years following the introduction by Schramm [Sch00] of random curves called Schramm-Loewner evolution (SLE). This approach, centred around the geometric description of critical models of statistical physics, has led to exact statements on the interfaces of percolation or the critical Ising model; following the introduction of SLE and the work of Smirnov, probabilists also managed to justify and construct the CFT correlation functions of the scaling limit of the 2d Ising model [ChSm12,CHI15] (see also the review [Pel19] for the construction of CFT correlations via SLE observables).…”

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confidence: 99%

Conformal bootstrap in Liouville Theory

Guillarmou¹,

Kupiainen²,

Rhodes³

et al. 2020

Preprint

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Liouville conformal field theory (denoted LCFT) is a 2-dimensional conformal field theory depending on a parameter γ ∈ R and studied since the eighties in theoretical physics. In the case of the theory on the 2-sphere, physicists proposed closed formulae for the n-point correlation functions using symmetries and representation theory, called the DOZZ formula (for n = 3) and the conformal bootstrap (for n > 3). In a recent work, the three last authors introduced with F. David a probabilistic construction of LCFT for γ ∈ (0, 2] and proved the DOZZ formula for this construction. In this sequel work, we give the first mathematical proof that the probabilistic construction of LCFT on the 2-sphere is equivalent to the conformal bootstrap for γ ∈ (0, √ 2). Our proof combines the analysis of a natural semi-group, tools from scattering theory and the use of the Virasoro algebra in the context of the probabilistic approach (the so-called conformal Ward identities).

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Toward a conformal field theory for Schramm-Loewner evolutions

Cited by 13 publications

References 149 publications

UST branches, martingales, and multiple SLE(2)

UST branches, martingales, and multiple SLE(2)

Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=-2$ Logarithmic CFT

Conformal bootstrap in Liouville Theory

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