Schramm--Loewner Evolution in the Random Scatterer Henon-percolation Landscapes

Najafi, M. N.; Tizdast, S.; Cheraghalizadeh, J.

doi:10.5506/aphyspolb.50.929

Cited by 2 publications

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“…The importance of this theory is due to the fact that many properties of its traces are known. Examples of the predictions of SLE are the left passage probability (LPP) [3,4], the Fokker-Planck equation [5], the winding angle (WA) statistics [6,7], the crossing probability [1], the relation between the fractal dimension and κ [1,8], locality for κ = 6 and restriction for κ = 8 3 [1]. Despite of its power in characterizing models, the SLE theory is applicable only to systems fulfilling conformal invariance.…”

Section: Introductionmentioning

confidence: 99%

Self-similar but not conformally invariant traces obtained by modified Loewner forces

Tizdast,

Ebadi,

Cheraghalizadeh

et al. 2021

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The two-dimensional Loewner exploration process is generalized to the case where the random force is self-similar with positively correlated increments. We model this random force by a fractional Brownian motion with Hurst exponent H ≥ 1 2 ≡ HBM, where HBM stands for the one-dimensional Brownian motion. By manipulating the deterministic force, we design a scale-invariant equation describing self-similar traces which lack conformal invariance. The model is investigated in terms of the "input diffusivity parameter" κ, which coincides with the one of the ordinary Schramm-Loewner evolution (SLE) at H = HBM. In our numerical investigation, we focus on the scaling properties of the traces generated for κ = 2, 3, κ = 4 and κ = 6, 8 as the representatives, respectively, of the dilute phase, the transition point and the dense phase of the ordinary SLE. The resulting traces are shown to be scale-invariant. Using two equivalent schemes, we extract the fractal dimension, D f (H), of the traces which decrease monotonically with increasing H, reaching D f = 1 at H = 1 for all κ values. The left passage probability (LPP) test demonstrates that, for H values not far from the uncorrelated case (small H ≡ H−H BM H BM ) the prediction of the ordinary SLE is applicable with an effective diffusivity parameter κ eff . Not surprisingly, the κ eff 's do not fulfill the prediction of SLE for the relation between D f (H) and the diffusivity parameter.

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