Schramm-Loewner evolution (SLE)

Lawler, Gregory F.; Kozdron, Michael J.; Masson, Robert; Narayanan, Hariharan

doi:10.1090/pcms/016/05

Cited by 42 publications

(45 citation statements)

References 47 publications

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“…The following is proved in [6] (the proof there is in the upper half plane, but it immediately extends by conformal invariance).…”

Section: Schramm-loewner Evolution (Sle) and Notationmentioning

confidence: 92%

A natural parametrization for the Schramm–Loewner evolution

Lawler¹,

Sheffield⋆²

2011

Ann. Probab.

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The natural paramterization or length for the Schramm-Loewner evolution (SLE κ ) is the candidate for the scaling limit of the length of discrete curves for κ < 8. We improve the proof of the existence of the parametrization and use this to establish some new results. In particular, we show that the natural parametrization is independent of domain and it is Hölder continuous with respect to the capacity parametrization. We also give up-to-constants bounds for the two-point Green's function. Although we do not prove the conjecture that the natural length is given by the appropriate Minkowski content, we do prove that the corresponding expectations converge.

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“…The following is proved in [6] (the proof there is in the upper half plane, but it immediately extends by conformal invariance).…”

Section: Schramm-loewner Evolution (Sle) and Notationmentioning

confidence: 92%

A natural parametrization for the Schramm–Loewner evolution

Lawler¹,

Sheffield⋆²

2011

Ann. Probab.

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“…The scaling behaviour in the continuum limit of many 2d stochastic processes, as critical percolation, the critical Ising model, self-avoiding random walks, etc., can be described by the Schramm-Loewner evolution (known also as stochastic Loewner evolution or SLE). An SLE with parameter κ, or SLE κ , is essentially a family of conformally invariant random planar curves [21,22,23], with the parameter κ controlling how much the curve "turns". It has been shown [16,24] that the parameter κ is the same as the one in the aforementioned Coulomb gas formulation of the q-state Potts model, and it is linked to the central charge c of the associated conformal field theory: for example, SLE 3 describes the critical Ising model, SLE 6 describes critical percolation, etc.…”

Section: Observablesmentioning

confidence: 99%

Coarsening and percolation in the kinetic 2d Ising model with spin exchange updates and the voter model

Tartaglia¹,

Cugliandolo²,

Picco³

2018

J. Stat. Mech.

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We study the early time dynamics of bimodal spin systems on 2d lattices evolving with different microscopic stochastic updates. We treat the ferromagnetic Ising model with locally conserved order parameter (Kawasaki dynamics), the same model with globally conserved order parameter (nonlocal spin exchanges), and the voter model. As already observed for non-conserved order parameter dynamics (Glauber dynamics), in all the cases in which the stochastic dynamics satisfy detailed balance, the critical percolation state persists over a long period of time before usual coarsening of domains takes over and eventually takes the system to equilibrium. By studying the geometrical and statistical properties of time-evolving spin clusters we are able to identify a characteristic length p (t), different from the usual length d (t) ∼ t 1/z d that describes the late time coarsening, that is involved in all scaling relations in the approach to the critical percolation regime. We find that this characteristic length depends on the particular microscopic dynamics and the lattice geometry. In the case of the voter model, we observe that the system briefly passes through a critical percolation state, to later approach a dynamical regime in which the scaling behaviour of the geometrical properties of the ordered domains can be ascribed to a different criticality.arXiv:1805.05775v2 [cond-mat.stat-mech]

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“…In the case n = 2, the right-hand side is comparable to a sharp estimate of the 2-point Green's function given in [7] up to a constant. Thus, we expect that it holds for all n ∈ N. 4. We guess that one can show E[e λCont d (γ∩D) ] < ∞ for some λ > 0 in any bounded domain D. This is nice because we can study natural length by its moment generating function.…”

Section: Introductionmentioning

confidence: 96%