UST branches, martingales, and multiple SLE(2)

Karrila, Alex

doi:10.1214/20-ejp485

Cited by 10 publications

(4 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For N = 1, the formula for the unique connectivity weight is (22); for N = 2, the formula is (24); for N = 3, these formulas are ( 28) and (32); and for N = 4, these formulas are (35), (39) and (46). In addition, we find an explicit formula (56) for the 'rainbow connectivity weight' (section 2.5), generating a multiple-SLE κ process in which the boundary arcs join the 2N marked points on the real axis to form a 'rainbow' [4,42] in the upper half-plane (figure 19). (Reference [35] obtains an alternative explicit formula for the rainbow connectivity weight using the 'spin-chain Coulomb gas correspondence,' previously developed in [34].)…”

Section: Discussionmentioning

confidence: 76%

“…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%

See 1 more Smart Citation

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

Section: Discussionmentioning

confidence: 76%

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

show abstract

“…Next, define τ δn to be the first time when η δn 1 hits (x δn 2 x δn 2N ) and τ to be the first time when η 1 hits (x 2 x 2N ). By properly adjusting the coupling (as in [HLW20, Proof of Theorem 4.2, Equation (4.25)]; see also [Kar20,GW20]), we may assume lim n→∞ τ δn = τ almost surely. Now, denote by (W t , t ≥ 0) the driving function of η1 and by (g t , t ≥ 0) the corresponding conformal maps.…”

Section: Convergence Of the Observablementioning

confidence: 99%

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

Liu,

Peltola,

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

“…Dmitry Chelkak is grateful to Stanislav Smirnov for explaining the ideas of [24] during several conversations dating back to 2009-2014. We want to thank Michel Bauer, Konstantin Izyurov and Kalle Kytölä for valuable comments and for encouraging us to write this paper; Alex Karrila for useful discussions of his research [13,14]; Chengyang Shao for discussions during his spring 2017 internship at the ENS [31]; and Mikhail Skopenkov for a feedback, which in particular included pointing out a mess at the end of the proof of [8, Theorem 3.13]. This proof is sketched in Section 3 with a necessary correction.…”

mentioning

confidence: 99%

On the convergence of massive loop-erased random walks to massive SLE(2) curves

Chelkak

Wan

2021

Electron. J. Probab.

View full text Add to dashboard Cite

Following the strategy proposed by Makarov and Smirnov [24] in 2009 (see also [3,2] for theoretical physics arguments), we provide technical details for the proof of convergence of massive loop-erased random walks to the chordal mSLE(2) process. As no follow-up to [24] appeared since then, we believe that such a treatment might be of interest to the community. We do not require any regularity of the limiting planar domain near its degenerate prime ends a and b except that (Ω δ , a δ , b δ ) are assumed to be 'close discrete approximations' to (Ω, a, b) near a and b in the sense of [13].

show abstract

UST branches, martingales, and multiple SLE(2)

Cited by 10 publications

References 29 publications

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

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

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

On the convergence of massive loop-erased random walks to massive SLE(2) curves

Contact Info

Product

Resources

About

UST branches, martingales, and multiple SLE(2)

Cited by 10 publications

References 29 publications

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

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

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

On the convergence of massive loop-erased random walks to massive SLE(2) curves

Contact Info

Product

Resources

About

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

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