Schramm–Loewner evolution of the accessible perimeter of isoheight lines of correlated landscapes

2018 ◽  
Vol 29 (01) ◽  
pp. 1850008 ◽  
Author(s):  
N. Posé ◽  
K. J. Schrenk ◽  
N. A. M. Araújo ◽  
H. J. Herrmann
Keyword(s):  
Fractal Dimension ◽  
Random Walk ◽  
Long Range ◽  
Hurst Exponent ◽  
Diffusion Constant ◽  
Strongly Correlated ◽  
Loewner Evolution ◽  
Positive Formula ◽  
Practical Implications ◽  
Negative Formula

Real landscapes exhibit long-range height–height correlations, which are quantified by the Hurst exponent [Formula: see text]. We give evidence that for negative [Formula: see text], in spite of the long-range nature of correlations, the statistics of the accessible perimeter of isoheight lines is compatible with Schramm–Loewner evolution curves and therefore can be mapped to random walks, their fractal dimension determining the diffusion constant. Analytic results are recovered for [Formula: see text] and [Formula: see text] and a conjecture is proposed for the values in between. By contrast, for positive [Formula: see text], we find that the random walk is not Markovian but strongly correlated in time. Theoretical and practical implications are discussed.

scholarly journals Coset construction of Virasoro minimal models and coupling of Wess–Zumino–Witten theory with Schramm–Loewner evolution

2019 ◽  
Vol 31 (10) ◽  
pp. 1950037
Author(s):  
Shinji Koshida
Keyword(s):  
Field Theories ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Field ◽  
Loewner Evolution ◽  
Witten Theory ◽  
Coset Construction ◽  
New Perspective ◽  
Group Symmetries ◽  
Math Phys

Schramm–Loewner evolution (SLE) is a random process that gives a useful description of fractal curves. After its introduction, many works concerning the connection between SLE and conformal field theory (CFT) have been carried out. In this paper, we develop a new method of coupling SLE with a Wess–Zumino–Witten (WZW) model for [Formula: see text], an example of CFT, relying on a coset construction of Virasoro minimal models. Generalizations of SLE that correspond to WZW models were proposed by previous works [E. Bettelheim et al., Stochastic Loewner evolution for conformal field theories with Lie group symmetries, Phys. Rev. Lett. 95 (2005) 251601] and [Alekseev et al., On SLE martingales in boundary WZW models, Lett. Math. Phys. 97 (2011) 243–261], in which the parameters in the generalized SLE for [Formula: see text] were related to the level of the corresponding [Formula: see text]-WZW model. The present work unveils the mechanism of how the parameters were chosen, and gives a simpler proof of the result in these previous works, shedding light on a new perspective of SLE/WZW coupling.

scholarly journals Schramm–Loewner evolution with Lie superalgebra symmetry

2018 ◽  
Vol 33 (20) ◽  
pp. 1850117 ◽  
Author(s):  
Shinji Koshida
Keyword(s):  
Differential Equations ◽  
Representation Theory ◽  
Stochastic Differential Equations ◽  
Degrees Of Freedom ◽  
Lie Superalgebra ◽  
Classical Type ◽  
Finite Dimensional ◽  
Loewner Evolution ◽  
Internal Degrees Of Freedom

We propose a generalization of Schramm–Loewner evolution (SLE) that has internal degrees of freedom described by an affine Lie superalgebra. We give a general formulation of SLE corresponding to representation theory of an affine Lie superalgebra whose underlying finite-dimensional Lie superalgebra is basic classical type, and write down stochastic differential equations on internal degrees of freedom in case that the corresponding affine Lie superalgebra is [Formula: see text]. We also demonstrate computation of local martingales associated with the solution from a representation of [Formula: see text].

CONFORMAL INVARIANCE OF CONTOUR LINES ON THE (2 + 1)-DIMENSIONAL RESTRICTED SOLID-ON-SOLID SURFACE

2011 ◽  
Vol 25 (04) ◽  
pp. 255-264 ◽  
Author(s):  
WEI ZHOU ◽  
GANG TANG ◽  
KUI HAN ◽  
HUI XIA ◽  
DA-PENG HAO ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Solid Surface ◽  
Large Scale ◽  
Fractal Dimensions ◽  
Universality Class ◽  
Invariant Curves ◽  
Conformal Invariant ◽  
Contour Lines ◽  
Loewner Evolution ◽  
Scale Limit

The contour lines of the saturated surface of the (2 + 1)-dimensional restricted solid-on-solid (RSOS) growth model are investigated by numerical method. It is shown that the calculated contour lines are conformal invariant curves with fractal dimension df = 1.34, and they belong to the universality class at large-scale limit, called the Schramm–Loewner evolution with diffusivity κ = 4. This is identical to the value obtained from the inverse cascade of surface quasigeostrophic (SQG) turbulence [Phys. Rev. Lett.98 (2007) 024501]. We also found that the measured fractal dimensions of contours on the (2 + 1)-dimensional RSOS saturated surfaces do not coincide well with that of SLE4 df = 1 + κ/8.

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