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

Liu, Mingchang; Peltola, Eveliina; Wu, Hao

doi:10.48550/arxiv.2108.04421

Cited by 4 publications

(5 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the physics literature, percolation is usually investigated as a (non-unitary) conformal field theory (CFT) with central charge 𝑐 = 0. It is also a prototypical example of a logarithmic field theory [26,61], so that its study belongs to an area of research very active both in physics and mathematics (see, e.g., [26] and [47]). As such, percolation is often studied by analyzing connection probabilities (sometimes called connectivity functions).…”

Section: Background and Motivationmentioning

confidence: 99%

Conformal covariance of connection probabilities and fields in 2D critical percolation

Camia

2023

Comm Pure Appl Math

View full text Add to dashboard Cite

Fitting percolation into the conformal field theory framework requires showing that connection probabilities have a conformally invariant scaling limit. For critical site percolation on the triangular lattice, we prove that the probability that n vertices belong to the same open cluster has a well‐defined scaling limit for every . Moreover, the limiting functions transform covariantly under Möbius transformations of the plane as well as under local conformal maps, that is, they behave like correlation functions of primary operators in conformal field theory. In particular, they are invariant under translations, rotations and inversions, and for any . This implies that and , for some constants C2 and C3.We also define a site‐diluted spin model whose n‐point correlation functions Cn can be expressed in terms of percolation connection probabilities and, as a consequence, have a well‐defined scaling limit with the same properties as the functions . In particular, . We prove that the magnetization field associated with this spin model has a well‐defined scaling limit in an appropriate space of distributions. The limiting field transforms covariantly under Möbius transformations with exponent (scaling dimension) 5/48. A heuristic analysis of the four‐point function of the magnetization field suggests the presence of an additional conformal field of scaling dimension 5/4, which counts the number of percolation four‐arm events and can be identified with the so‐called “four‐leg operator” of conformal field theory.

show abstract

Section: Background and Motivationmentioning

confidence: 99%

Conformal covariance of connection probabilities and fields in 2D critical percolation

Camia

2023

Comm Pure Appl Math

View full text Add to dashboard Cite

show abstract

“…In general, for the range κ ∈ (0, 8], to our knowledge there are explicit formulas for Z α only when κ / ∈ Q (cf. [KP16]) and for a few special rational cases: κ = 2 [KKP20]; κ = 4 [PW19]; and κ = 8 [LPW21]. For κ ∈ (0, 6], an explicit probabilistic construction was given in [Wu20, Theorem 1.7], which immediately implies (1.15).…”

Section: Conjectures For Random-cluster Modelsmentioning

confidence: 99%

“…Its loop representation contains N interfaces η δ 2r−1 starting from y δ, 2r−1 , with 1 ≤ r ≤ N , terminating among the medial vertices {y δ, 2r : 1 ≤ r ≤ N }. Inspired by [LPW21] (see also [Izy15, Figure 2]), we define an exploration path ξ δ starting from the outer corner y δ, 1 and terminating at the outer corner y δ, 2 via the following procedure (see Figure 3.1).…”

Section: Exploration Process and Holomorphic Spinor Observablementioning

confidence: 99%

Connection Probabilities of Multiple FK-Ising Interfaces

Ye¹,

Peltola²,

Wu³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We find the scaling limits of a general class of boundary-to-boundary connection probabilities and multiple interfaces in the critical planar FK-Ising model, thus verifying predictions from the physics literature. We also discuss conjectural formulas using Coulomb-gas integrals for the corresponding quantities in general critical planar random-cluster models with cluster-weight q ∈ [1, 4). Thus far, proofs for convergence, including ours, rely on discrete complex analysis techniques and are beyond reach for other values of q than the FK-Ising model (q = 2). Given the convergence of interfaces, the conjectural formulas for other values of q could be verified similarly with relatively minor technical work. The limit interfaces are variants of SLE κ curves (with κ = 16/3 for q = 2). Their partition functions, that give the connection probabilities, also satisfy properties predicted for correlation functions in conformal field theory (CFT), expected to describe scaling limits of critical random-cluster models. We verify these properties for all q ∈ [1, 4), thus providing further evidence of the expected CFT description of these models.

show abstract

“…Our results show that such expressions may reduce to fewer terms, sometimes even a single term, for small N. Further, obtaining results for rational values of κ with quantum group methods is possible by taking the limit of the quantum group results but involves many cancelling terms, and thus appears complicated. See [52][53][54][55][56][57][58][59][60][61][62][63] for these and related results from quantum group and other methods.…”

Section: Introductionmentioning

confidence: 99%

Multiple-SLE_κ connectivity weights for rectangles, hexagons, and octagons

Flores¹,

Simmons²,

Kleban³

2022

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

In previous work, two of the authors determined, completely and rigorously, a solution space SN for a homogeneous system of 2N + 3 linear partial differential equations (PDEs) in 2N variables that arises in conformal field theory (CFT) and multiple Schramm-Lo ̈wner evolution (SLEκ). The system comprises 2N null-state equations and three conformal Ward identities that govern CFT correlation functions of 2N one-leg boundary operators or SLEκ partition functions. M. Bauer et al. conjectured a formula, expressed in terms of “pure SLEκ partition functions,” for the probability that the growing curves of a multiple-SLEκ process join in a particular connectivity. In a previous article, we rigorously define certain elements of SN , which we call “connectivity weights,” argue that they are in fact pure SLEκ partition functions, and show how to find explicit formulas for them in terms of Coulomb gas contour integrals. Our formal definition of the connectivity weights immediately leads to a method for finding explicit expressions for them. However, this method gives very complicated formulas where simpler versions may be available, and it is not applicable for certain values of κ ∈ (0, 8) corresponding to well-known critical lattice models in statistical mechanics. In this article, we determine expressions for all connectivity weights in SN for N ∈ {1, 2, 3, 4} (those with N ∈ {3, 4} are new) and for so-called “rainbow connectivity weights” in SN for all N ∈ Z+ + 1. We verify these formulas by explicitly showing that they satisfy the formal definition of a connectivity weight. In appendix B, we investigate logarithmic singularities of some of these expressions, appearing for certain values of κ predicted by logarithmic CFT. Keywords: conformal field theory, Schramm-Lo ̈wner evolution, connectivity weights, crossing probability, pure SLEκ partition function

show abstract

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

Cited by 4 publications

References 41 publications

Conformal covariance of connection probabilities and fields in 2D critical percolation

Conformal covariance of connection probabilities and fields in 2D critical percolation

Connection Probabilities of Multiple FK-Ising Interfaces

Multiple-SLE_κ connectivity weights for rectangles, hexagons, and octagons

Contact Info

Product

Resources

About

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

Cited by 4 publications

References 41 publications

Conformal covariance of connection probabilities and fields in 2D critical percolation

Conformal covariance of connection probabilities and fields in 2D critical percolation

Connection Probabilities of Multiple FK-Ising Interfaces

Multiple-SLE κ connectivity weights for rectangles, hexagons, and octagons

Contact Info

Product

Resources

About

Multiple-SLE_κ connectivity weights for rectangles, hexagons, and octagons