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

Chelkak, Dmitry; Wan, Yijun

doi:10.1214/21-ejp615

Cited by 8 publications

(3 citation statements)

References 25 publications

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“…This notion was introduced by Karrila in [Kar18], see in particular [Kar18, Theorem 4.2]. (See also [Kar19] and [CW21]. )…”

Section: Preliminaries On Ising Modelmentioning

confidence: 99%

Multiple Ising interfaces in annulus and $2N$-sided radial SLE

Ye¹,

Wu²,

Yang³

2023

Preprint

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We consider critical planar Ising model in annulus with alternating boundary conditions on the outer boundary and free boundary conditions in the inner boundary. As the size of the inner hole goes to zero, the event that all interfaces get close to the inner hole before they meet each other is a rare event. We prove that the law of the collection of the interfaces conditional on this rare event converges in total variation distance to the so-called 2N -sided radial SLE 3 , introduced by [HL21]. The proof relies crucially on an estimate for multiple chordal SLE. Suppose (γ 1 , . . . , γ N ) is chordal N -SLE κ with κ ∈ (0, 4] in the unit disc, and we consider the probability that all N curves get close to the origin. We prove that the limit lim r→0+ r −A 2N P[dist(0, γ j ) < r, 1 ≤ j ≤ N ] exists, where A 2N is the so-called 2N -arm exponents and dist is Euclidean distance. We call the limit Green's function for chordal N -SLE κ . This estimate is a generalization of previous conclusions with N = 1 and N = 2 proved in [LR12, LR15] and [Zha20] respectively.

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“…This notion was introduced by Karrila in [Kar18], see in particular [Kar18, Theorem 4.2]. (See also [Kar19] and [CW21]. )…”

Section: Preliminaries On Ising Modelmentioning

confidence: 99%

Multiple Ising interfaces in annulus and $2N$-sided radial SLE

Ye¹,

Wu²,

Yang³

2023

Preprint

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“…The following notion was introduced by Karrila in [Kar18], see in particular [Kar18, Theorem 4.2]. (See also [Kar19] and [CW21].) Definition 3.1.…”

Section: Convergence Of Polygonsmentioning

confidence: 99%

Connection Probabilities of Multiple FK-Ising Interfaces

Ye¹,

Peltola²,

Wu³

2022

Preprint

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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.

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“…Martingale arguments proving the convergence lattice interfaces to different multiple SLE type curves have been earlier given in [Izy17, KS19,KS18,Kar19]. The convergence of different variants of a single UST branch, or the closely related loop-erased random walk, to different SLE variants has been proven in [LSW04,Zha08,LV16,CW19], and for isoradial and even more general graphs in [CS11, YY11, Suz14, Uch17].…”

mentioning

confidence: 99%

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

Cited by 8 publications

References 25 publications

Multiple Ising interfaces in annulus and $2N$-sided radial SLE

Multiple Ising interfaces in annulus and $2N$-sided radial SLE

Connection Probabilities of Multiple FK-Ising Interfaces

UST branches, martingales, and multiple SLE(2)

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